## 5 Steps to a 5: AP Physics C (2016)

### STEP __4__

### Review the Knowledge You Need to Score High

### CHAPTER 20

### Magnetism

**IN THIS CHAPTER**

**Summary:** Magnetic fields produce forces on moving charges; moving charges, such as current-carrying wires, can create magnetic fields. This chapter discusses the production and the effects of magnetic fields.

**Key Ideas**

The force on a moving charge due to a magnetic field is *qvB* .

The direction of the magnetic force on a moving charge is given by a right-hand rule, and is NOT in the direction of the magnetic field.

Current-carrying wires produce magnetic fields.

When the magnetic flux through a wire changes, a voltage is induced.

An inductor inhibits the change in the current running through it. After a long time, the inductor acts as a bare wire.

**Relevant Equations**

Force on a charged particle in a magnetic field:

*F* = *qvB*

Force on a current-carrying wire:

*F* = *ILB*

Magnetic field due to a long, straight, current-carrying wire:

Magnetic flux:

*Φ* _{B}_{ }= *BA*

Induced EMF:

Induced EMF for a rectangular wire moving into or out of a magnetic field:

*ε* = *BLv*

Time constant for an LR circuit:

When most people think of magnets, they imagine horseshoe-shaped objects that can pick up bits of metal. Or maybe they visualize a refrigerator door. But not a physics ace like you! You know that magnetism is a wildly diverse topic, involving everything from bar magnets to metal coils to mass spectrometers. Perhaps you also know that magnetism is a subject filled with countless “right-hand rules,” many of which can seem difficult to use or just downright confusing. So our goal in this chapter—besides reviewing all of the essential concepts and formulas that pertain to magnetism—is to give you a set of easy-to-understand, easy-to-use right-hand rules that are guaranteed to earn you points on the AP exam.

**Magnetic Fields**

All magnets are dipoles, which means that they have two “poles,” or ends. One is called the north pole, and the other is the south pole. Opposite poles attract, and like poles repel.

You can never create a magnet with just a north pole or just a south pole. If you took the magnet in __Figure 20.1__

**Figure 20.1 Bar magnet.**

and cut it down the middle, you would not separate the poles. Instead, you would create two magnets like those shown in __Figure 20.2__ .

**Figure 20.2 Cutting the bar magnet in Figure 20.1 in half just gives you two smaller bar magnets. You can never get an isolated north or south pole.**

A magnet creates a magnetic field. (See __Figure 20.3__ .) Unlike electric field lines, which either go from a positive charge to a negative charge or extend infinitely into space, magnetic field lines form loops. These loops point away from the north end of a magnet, and toward the south end. Near the magnet the lines point nearly straight into or out of the pole.

**Figure 20.3 Magnetic field lines created by a bar magnet.**

Just as we talk about the value of an electric field at a certain point, we can also talk about the value of a magnetic field at a certain point. The value of a magnetic field is a vector quantity, and it is abbreviated with the letter *B* . The value of a magnetic field is measured in teslas.

Often, the writers of the AP exam like to get funky about how they draw magnetic field lines. Rather than putting a magnetic field in the plane of the page, so that the field would point up or down or left or right, the AP writers will put magnetic fields perpendicular to the page. This means that the magnetic field either shoots out toward you or shoots down into the page.

When a magnetic field line is directed out of the page, it is drawn as shown in __Figure 20.4a__ ,

**Figure 20.4a Symbol for a magnetic field line directed out of the page.**

and when a magnetic field line is directed into the page, it is drawn as shown in __Figure 20.4b__ .

**Figure 20.4b Symbol for a magnetic field line directed into the page.**

Supposedly, the drawing in __Figure 20.4a__ is intended to look like the tip of an arrow coming out of a page, and the drawing in __Figure 20.4b__ is intended to look like the tail of an arrow going into a page.^{ }^{1}^{ }These symbols can be used to describe other ideas, such as electric fields going into or out of the page, or currents flowing into or out of the page, but they are most often used to describe magnetic fields.

**Long, Straight, Current-Carrying Wires**

Bar magnets aren”t the only things that create magnetic fields—current-carrying wires do also. Of course, you can also create a magnetic field using a short, curvy, current-carrying wire, but the equations that describe that situation are a little more complicated, so we”ll focus on long, straight, current-carrying wires.

The magnetic field created by a long, straight, current-carrying wire loops around the wire in concentric circles. The direction in which the magnetic field lines loop is determined by a right-hand rule.

(Incidentally, our versions of the right-hand rules may not be the same as what you”ve learned in physics class. If you”re happy with the ones you already know, you should ignore our advice and just stick with what works best for you.)

Right-hand rule: To find the direction of the B field produced by long, straight, current-carrying wires.

Pretend you are holding the wire with your right hand. Point your thumb in the direction of the current. Your fingers wrap around your thumb the same way that the magnetic field wraps around the wire.

Here”s an example. A wire is directed perpendicular to the plane of this page (that is, it”s coming out straight toward you). The current in this wire is flowing out of the page. What does the magnetic field look like?

To solve this, we first pretend that we are grabbing the wire. If it helps, take your pencil and place it on this page, with the eraser touching the page and the point of the pencil coming out toward you. This pencil is like the wire. Now grab the pencil with your right hand. The current is coming out of the page, so make sure that you have grabbed the pencil in such a way that your thumb is pointing away from the page. If it looks like you”re giving someone a “thumbs-up sign,” then you”re doing this correctly. Finally, look at how your fingers are wrapped around the pencil. From a bird”s-eye view, it should look like your fingers are wrapping counterclockwise. So this tells us the answer to the problem, as shown in __Figure 20.5__ .

**Figure 20.5 Magnetic field (dotted lines) generated by a long, straight, current-carrying wire oriented perpendicular to the plane of the page.**

Here”s another example. What does the magnetic field look like around a wire in the plane of the page with current directed upward?

We won”t walk you through this one; just use the right-hand rule, and you”ll be fine. The answer is shown in __Figure 20.6__ .

**Figure 20.6 Magnetic field around a wire in the plane of the page with current directed upward.**

The formula that describes the magnitude of the magnetic field created by a long, straight, current-carrying wire is the following:

In this formula, *B* is the magnitude of the magnetic field, μ_{0} is a constant called the “permeability of free space” (μ_{0} = 4π × 10^{−7} T·m/A), *I* is the current flowing in the wire, and *r* is the distance from the wire.

**Moving Charged Particles**

The whole point of defining a magnetic field is to determine the forces produced on an object by the field. You are familiar with the forces produced by bar magnets—like poles repel, opposite poles attract. We don”t have any formulas for the amount of force produced in this case, but that”s okay, because this kind of force is irrelevant to the AP exam.

Instead, we must focus on the forces produced by magnetic fields on charged particles, including both isolated charges and current-carrying wires. (After all, current is just the movement of positive charges.)

A magnetic field exerts a force on a charged particle if that particle is moving perpendicular to the magnetic field. A magnetic field does not exert a force on a stationary charged particle, nor on a particle that is moving parallel to the magnetic field.

The magnitude of the force exerted on the particle equals the charge on the particle, *q* , multiplied by the velocity of the particle, *v* , multiplied by the magnitude of the magnetic field.

This equation is sometimes written as *F* = *qvB* (sin *θ* ). The *θ* refers to the angle formed between the velocity vector of your particle and the direction of the magnetic field. So, if a particle moves in the same direction as the magnetic field lines, *θ* = 0°, sin 0° = 0, and *that particle experiences no magnetic force!*

Nine times out of ten, you will not need to worry about this “sin *θ* ” term, because the angle will either be zero or 90°. However, if a problem explicitly tells you that your particle is *not* traveling perpendicular to the magnetic field, then you will need to throw in this extra “sin *θ* ” term.

Right-hand rule: To find the force on a charged particle.

Point your right hand, with fingers extended, in the direction that the charged particle is traveling. Then, bend your fingers so that they point in the direction of the magnetic field.

- If the particle has a POSITIVE charge, your thumb points in the direction of the force exerted on it.
- If the particle has a NEGATIVE charge, your thumb points opposite the direction of the force exerted on it.

The key to this right-hand rule is to remember the sign of your particle. This next problem illustrates how important sign can be.

An electron travels through a magnetic field, as shown below. The particle”s initial velocity is 5 × 10^{6} m/s, and the magnitude of the magnetic field is 0.4 T. What are the magnitude and direction of the particle”s acceleration?

This is one of those problems where you”re told that the particle is *not* moving perpendicular to the magnetic field. So the formula we use to find the magnitude of the force acting on the particle is

*F* = *qvB* (sin *θ* )

*F* = (1.6 × 10^{−19} C)(5 × 10^{6} m/s)(0.4 T)(sin 30°)

*F* = 1.6 × 10^{−13} N.

Note that we never plug in the negative signs when calculating force. The negative charge on an electron will influence the direction of the force, which we will determine in a moment. Now we solve for acceleration:

Wow, you say … a bigger acceleration than anything we”ve ever dealt with. Is this unreasonable? After all, in less than a second the particle would be moving faster than the speed of light, right? The answer is still reasonable. In this case, the acceleration is perpendicular to the velocity. This means the acceleration is *centripetal* , and the particle must move in a circle at constant speed. But even if the particle were speeding up at this rate, either the acceleration wouldn”t act for very long, or relativistic effects would prevent the particle from traveling faster than light.

Finally, we solve for direction using the right-hand rule. We point our hand in the direction that the particle is traveling—to the right. Next, we curl our fingers upward, so that they point in the same direction as the magnetic field. Our thumb points out of the page. BUT WAIT!!! We”re dealing with an electron, which has a negative charge. So the force acting on our particle, and therefore the particle”s acceleration, points in the opposite direction. The particle is accelerating into the page.

**Magnetic Force on a Wire**

A current is simply the flow of positive charges. So, if we put a current-carrying wire perpendicular to a magnetic field, we have placed moving charges perpendicular to the field, and these charges experience a force. The wire can be pulled by the magnetic field!

The formula for the force on a long, straight, current-carrying wire in the presence of a magnetic field is

This equation says that the force on a wire equals the current in the wire, *I* , multiplied by the length of the wire, *L* , multiplied by the magnitude of the magnetic field, *B* , in which the wire is located.

Sometimes you”ll see this equation written as *F* = *ILB* (sin *θ* ). Just like the equation for the force on a charge, the *θ* refers to the angle between the wire and the magnetic field. You normally don”t have to worry about this *θ*because, in most problems, the wire is perpendicular to the magnetic field, and sin 90° = 1, so the term cancels out.

The direction of the force on a current-carrying wire is given by the same right-hand rule as for the force on a charged particle because current is simply the flow of positive charge.

What would happen if you had two long, straight, current-carrying wires side by side? This is a question that the writers of the AP exam love to ask, so it is a great idea to learn how to answer it.

The trick that makes answering this question very easy is that you have to draw the direction of the magnetic field that one of the wires creates; then consider the force on the other wire. So, for example …

Two wires are placed parallel to each other. The direction of current in each wire is indicated above. How will these wires interact?

(A) They will attract each other.

(B) They will repel each other.

(C) They will not affect each other.

(D) This question cannot be answered without knowing the length of each wire.

(E) This question cannot be answered without knowing the current in each wire.

Let”s follow our advice and draw the magnetic field created by the left-hand wire.

Now, a wire”s field cannot produce a force on itself. The field that we drew is *caused by* the left wire, but produces a force on the right-hand wire. Which direction is that force? Use the right-hand rule for the force on a charged particle. The charges are moving up, in the direction of the current. So point up the page, and curl your fingers toward the magnetic field, into the page. The right wire is forced to the LEFT. Newton”s third law says that the force on the left wire by the right wire will be equal and opposite.^{ }^{2}^{ }So, the wires attract, answer A.

Often, textbooks give you advice such as, “Whenever the current in two parallel wires is traveling in the same direction, the wires will attract each other, and vice versa.” Use it if you like, but this advice can easily be confused.

**Mass Spectrometry: More Charges Moving Through Magnetic Fields**

A magnetic field can make a charged particle travel in a circle. Here”s how it performs this trick.

**Figure 20.7a Positively charged particle moving in a magnetic field directed out of the page.**

Let”s say you have a proton traveling through a uniform magnetic field coming out of the page, and the proton is moving to the right, like the one we drew in __Figure 20.7a__ . The magnetic field exerts a downward force on the particle (use the right-hand rule). So the path of the particle begins to bend downward, as shown in __Figure 20.7b__ .

**Figure 20.7b Curving path of a positively charged particle moving in a magnetic field directed out of the page.**

Now our proton is moving straight down. The force exerted on it by the magnetic field, using the right-hand rule, is now directed to the left. So the proton will begin to bend leftward. You probably see where this is going—a charged particle, traveling perpendicular to a uniform magnetic field, will follow a circular path.

We can figure out the radius of this path with some basic math. The force of the magnetic field is causing the particle to go in a circle, so this force must cause centripetal acceleration. That is, *qvB* = *mv* ^{2} /*r* .

We didn”t include the “sin *θ* ” term because the particle is always traveling perpendicular to the magnetic field. We can now solve for the radius of the particle”s path:

The real-world application of this particle-in-a-circle trick is called a mass spectrometer. A mass spectrometer is a device used to determine the mass of a particle.

A mass spectrometer, in simplified form, is drawn in __Figure 20.8__ .

**Figure 20.8 Basic mass spectrometer.**

A charged particle enters a uniform electric field (shown at the left in __Figure 20.8__ ). It is accelerated by the electric field. By the time it gets to the end of the electric field, it has acquired a high velocity, which can be calculated using conservation of energy. Then the particle travels through a tiny opening and enters a uniform magnetic field. This magnetic field exerts a force on the particle, and the particle begins to travel in a circle. It eventually hits the wall that divides the electric-field region from the magnetic-field region. By measuring where on the wall it hits, you can determine the radius of the particle”s path. Plugging this value into the equation we derived for the radius of the path, you can calculate the particle”s mass *r* = *mv/qB* .

You may see a problem on the free-response section that involves a mass spectrometer. These problems may seem intimidating, but, when you take them one step at a time, they”re not very difficult.

**Induced EMF**

A changing magnetic field produces a current. We call this occurrence **electromagnetic induction** .

So let”s say that you have a loop of wire in a magnetic field. Under normal conditions, no current flows in your wire loop. However, if you change the magnitude of the magnetic field, a current will begin to flow.

We”ve said in the past that current flows in a circuit (and a wire loop qualifies as a circuit, albeit a simple one) when there is a potential difference between the two ends of the circuit. Usually, we need a battery to create this potential difference. But we don”t have a battery hooked up to our loop of wire. Instead, the changing magnetic field is doing the same thing as a battery would. So rather than talking about the voltage of the battery in this circuit, we talk about the “voltage” created by the changing magnetic field. The technical term for this “voltage” is **induced EMF** .

**Induced EMF** : The potential difference created by a changing magnetic field that causes a current to flow in a wire. EMF stands for Electro-Motive Force, but is *NOT* a force.

For a loop of wire to “feel” the changing magnetic field, some of the field lines need to pass through it. The amount of magnetic field that passes through the loop is called the **magnetic flux** . This concept is pretty similar to electric flux.

**Magnetic Flux** : The number of magnetic field lines that pass through an area

The units of flux are called webers; 1 weber = 1 T·m^{2} . The equation for magnetic flux is

In this equation, *Φ* _{B}_{ }is the magnetic flux, *B* is the magnitude of the magnetic field, and *A* is the area of the region that is penetrated by the magnetic field.

Let”s take a circular loop of wire, lay it down on the page, and create a magnetic field that points to the right, as shown in __Figure 20.9__ .

**Figure 20.9 Loop of wire in the plane of a magnetic field.**

No field lines go through the loop. Rather, they all hit the edge of the loop, but none of them actually passes through the center of the loop. So we know that our flux should equal zero.

Okay, this time we will orient the field lines so that they pass through the middle of the loop. We”ll also specify the loop”s radius = 0.2 m, and that the magnetic field is that of the Earth, *B* = 5 × 10^{−5} T. This situation is shown in __Figure 20.10__ .

**Figure 20.10 Loop of wire with magnetic field lines going through it.**

Now all of the area of the loop is penetrated by the magnetic field, so *A* in the flux formula is just the area of the circle, π*r* ^{2} .

The flux here is

*Φ* _{B}_{ }= (5 × 10^{−5} )(π) (0.2^{2} ) = 6.2 × 10^{−6} T·m^{2} .

Sometimes you”ll see the flux equation written as *BA* cos*θ* . The additional cosine term is only relevant when a magnetic field penetrates a wire loop at some angle that”s not 90°. The angle *θ* is measured between the magnetic field and the “normal” to the loop of wire … if you didn”t get that last statement, don”t worry about it. Rather, know that the cosine term goes to 1 when the magnetic field penetrates directly into the loop, and the cosine term goes to zero when the magnetic field can”t penetrate the loop at all.

Because a loop will only “feel” a changing magnetic field if some of the field lines pass through the loop, we can more accurately say the following: *A changing magnetic **flux** creates an induced EMF* .

Faraday”s law tells us exactly how much EMF is induced by a changing magnetic flux.

ε is the induced EMF, *N* is the number of loops you have (in all of our examples, we”ve only had one loop), and Δ*t* is the time during which your magnetic flux, *Φ* _{B}_{ }, is changing.

Up until now, we”ve just said that a changing magnetic flux creates a current. We haven”t yet told you, though, in which direction that current flows. To do this, we”ll turn to **Lenz”s Law** .

**Lenz”s Law:** States that the direction of the induced current opposes the increase in flux

When a current flows through a loop, that current creates a magnetic field. So what Lenz said is that the current that is induced will flow in such a way that the magnetic field it creates points opposite to the direction in which the already existing magnetic flux is changing.

Sound confusing?^{ }^{4}^{ }It”ll help if we draw some good illustrations. So here is Lenz”s Law in pictures.

We”ll start with a loop of wire that is next to a region containing a magnetic field (__Figure 20.11a__ ). Initially, the magnetic flux through the loop is zero.

**Figure 20.11a Loop of wire next to a region containing a magnetic field pointing out of the page.**

Now, we will move the wire into the magnetic field. When we move the loop toward the right, the magnetic flux will increase as more and more field lines begin to pass through the loop. The magnetic flux is increasing out of the page—at first, there was no flux out of the page, but now there is some flux out of the page. Lenz”s Law says that the induced current will create a magnetic field that opposes this increase in flux. So the induced current will create a magnetic field into the page. By the right-hand rule, the current will flow clockwise. This situation is shown in __Figure 20.11b__ .

**Figure 20.11b Current induced in loop of wire as it moves into a magnetic field directed out of the page.**

After a while, the loop will be entirely in the region containing the magnetic field. Once it enters this region, there will no longer be a changing flux, because no matter where it is within the region, the same number of field lines will always be passing through the loop. Without a changing flux, there will be no induced EMF, so the current will stop. This is shown in __Figure 20.11c__ .

**Figure 20.11c Loop of wire with no current flowing, because it is not experiencing a changing magnetic flux.**

To solve a problem that involves Lenz”s Law, use this method:

- Point your right thumb in the initial direction of the magnetic field.
- Ask yourself, “Is the flux increasing or decreasing?”
- If the flux is decreasing, then just curl your fingers (with your thumb still pointed in the direction of the magnetic field). Your fingers show the direction of the induced current.
- If flux is increasing in the direction you”re pointing, then flux is decreasing in the other direction. So, point your thumb in the opposite direction of the magnetic field, and curl your fingers. Your fingers show the direction of the induced current.

**Induced EMF in a Rectangular Wire**

Consider the example in __Figures 20.11a__ –__c__ with the circular wire being pulled through the uniform magnetic field. It can be shown that if instead we pull a *rectangular* wire into or out of a uniform field *B* at constant speed *v* , then the induced EMF in the wire is found by

Here, *L* represents the length of the side of the rectangle that is NOT entering or exiting the field, as shown below in __Figure 20.12__ .

**Figure 20.12 Rectangular wire moving through a uniform magnetic field.**

**Some Words of Caution**

We say this from personal experience. First, when using a right-hand rule, use big, easy-to-see gestures. A right-hand rule is like a form of advertisement: it is a way that your hand tells your brain what the answer to a problem is. You want that advertisement to be like a billboard—big, legible, and impossible to misread. Tiny gestures will only lead to mistakes. Second, when using a right-hand rule, *always* use your right hand. *Never use your left hand!* This will cost you points!

**Exam tip from an AP Physics veteran:**

Especially if you hold your pencil in your right hand, it”s easy accidentally to use your left hand. Be careful!

—*Jessica, college sophomore*

**The Biot-Savart Law and Ampére”s Law**

So far we”ve only discussed two possible ways to create a magnetic field—use a bar magnet, or a long, straight, current-carrying wire. And of these, we only have an equation to find the magnitude of the field produced by the wire.

**Biot-Savart Law**

The Biot-Savart law provides a way, albeit a complicated way, to find the magnetic field produced by pretty much any type of current. It”s not worth worrying about using the law because it”s got a horrendously complicated integral with a cross product included. Just know the conceptual consequence: a little element of wire carrying a current produces a magnetic field that (a) wraps around the current element via the right-hand rule, and (b) decreases in magnitude as 1/*r* ^{2} , *r* being the distance from the current element.

So why does the magnetic field caused by a long, straight, current-carrying wire drop off as 1/*r* rather than 1/*r* ^{2} ? Because the 1/*r* ^{2} drop-off is for the magnetic field produced just by a teeny little bit of current-carrying wire (in calculus terminology, by a differential element of current). When we include the contributions of every teeny bit of a very long wire, the net field drops off as 1/*r* .

**Ampére”s Law**

Ampére”s law gives an alternative method for finding the magnetic field caused by a current. Although Ampére”s law is valid everywhere that current is continuous, it is only *useful* in a few specialized situations where symmetry is high. There are three important results of Ampére”s law:

- The magnetic field produced by a very long, straight current is

outside the wire; inside the wire, the field increases linearly from zero at the wire”s center.

- A solenoid is set of wound wire loops. A current-carrying solenoid produces a magnetic field. Ampére”s law can show that the magnetic field due to a solenoid is shaped like that of a bar magnet; and the magnitude of the magnetic field inside the solenoid is approximately uniform,
*B*= μ_{solenoid}_{0}*nI*. (Here*I*is the current in the solenoid, and*n*is the number of coils per meter in the solenoid.) - The magnetic field produced by a wire-wrapped torus (a “donut” with wire wrapped around it [see
__Figure 20.13__]) is zero everywhere outside the torus, but nonzero within the torus. The direction of the field inside the torus is around the donut.

**Figure 20.13 A wire-wrapped torus.**

**Maxwell”s Equations**

Okay, we”ll get this out of the way right now: *You will not have to solve Maxwell”s equations on the AP Physics exam* . These four equations include integrals the likes of which you will not be able to solve until well into college physics, if then. However, you *can* understand the basic point of each equation, and, most importantly, understand the equations” greatest consequence.

Accelerating charges produce oscillations of electric and magnetic fields. These oscillations propagate as waves, with speed

Maxwell obtained this wave speed as a mathematical result from the equations. He noticed that, when the experimentally determined constants were plugged in, the speed of his “electromagnetic waves” was identical to the speed of light.^{ }^{5}^{ }Maxwell”s conclusion was that light must be an electromagnetic wave.

What are Maxwell”s equations? We”re not even going to write them out, for fear that you might throw down your book in trepidation. If you”re really interested in the integral or differential form of the equations, you will find them in your physics book (or on a rather popular T-shirt). While we won”t write the equations, we”ll gladly summarize what they are and what they mean.

*Maxwell equation 1*is simply Gauss”s law: the net electric flux through a closed surface is proportional to the charge enclosed by that surface.*Maxwell equation 2*is sometimes called Gauss”s law for magnetism: the net magnetic flux through a closed surface must always be zero. The consequence of this equation is that magnetic poles come in north/south pairs—you cannot have an isolated north magnetic pole.*Maxwell equation 3*is simply Faraday”s law: a changing magnetic flux through a loop of wire induces an EMF.*Maxwell equation 4*is partly Ampére”s law, but with an addition called “displacement current” that allows the equation to be valid in all situations. The principal consequence is that just as a changing magnetic field can produce an electric field, a changing electric field can likewise produce a magnetic field.

** Practice Problems**

**Multiple Choice:**

** 1 .** A point charge of +1 μC moves with velocity

*v*into a uniform magnetic field

*B*directed to the right, as shown above. What is the direction of the magnetic force on the charge?

(A) to the right and up the page

(B) directly out of the page

(C) directly into the page

(D) to the right and into the page

(E) to the right and out of the page

** 2 .** A uniform magnetic field

*B*points up the page, as shown above. A loop of wire carrying a clockwise current is placed at rest in this field as shown above, and then let go. Which of the following describes the motion of the wire immediately after it is let go?

(A) The wire will expand slightly in all directions.

(B) The wire will contract slightly in all directions.

(C) The wire will rotate, with the top part coming out of the page.

(D) The wire will rotate, with the left part coming out of the page.

(E) The wire will rotate clockwise, remaining in the plane of the page.

** 3 .** An electron moves to the right in a uniform magnetic field that points into the page. What is the direction of the electric field that could be used to cause the electron to travel in a straight line?

(A) down toward the bottom of the page

(B) up toward the top of the page

(C) into the page

(D) out of the page

(E) to the left

**Free Response:**

** 4 .** A circular loop of wire of negligible resistance and radius

*R*= 20 cm is attached to the circuit shown above. Each resistor has resistance 10 Ω. The magnetic field of the Earth points up along the plane of the page in the direction shown, and has magnitude

*B*= 5.0 × 10

^{−5}T.

The wire loop rotates about a horizontal diameter, such that after a quarter rotation the loop is no longer in the page, but perpendicular to it. The loop makes 500 revolutions per second, and remains connected to the circuit the entire time.

(a) Determine the magnetic flux through the loop when the loop is in the orientation shown.

(b) Determine the maximum magnetic flux through the loop.

(c) Estimate the average value of the induced EMF in the loop.

(d) Estimate the average current through resistor *C* .

** 5 .** A loop of wire is located inside a uniform magnetic field, as shown above. Name at least four things you could do to induce a current in the loop.

** Solutions to Practice Problems**

__1__ .**C** —Use the right-hand rule for the force on charged particles. You point in the direction of the velocity, and curl your fingers in the direction of the magnetic field. This should get your thumb pointing into the page. Because this is a positive charge, no need to switch the direction of the force.

__2__ .**C** —Use the right-hand rule for the force on a wire. Look at each part of this wire. At the leftmost and rightmost points, the current is along the magnetic field lines. Thus, these parts of the wire experience no force. The topmost part of the wire experiences a force out of the page (point to the right, fingers curl up the page, the thumb points out of the page). The bottommost part of the wire experiences a force *into* the page. So, the wire will rotate.

__3__ .**A** —Use the right-hand rule for the force on a charge. Point in the direction of velocity, curl the fingers into the page, the thumb points up the page … but this is a *negative* charge, so the force on the charge is down the page. Now, the electric force must cancel the magnetic force for the charge to move in a straight line, so the electric force should be up the page. (E and B *fields* cannot cancel, but forces sure can.) The direction of an *electric* force on a negative charge is opposite the field; so the field should point down, toward the bottom of the page.

** 4 .** (a) Flux equals zero because the field points along the loop, not ever going straight through the loop.

(b) Flux is maximum when the field *is* pointing straight through the loop; that is, when the loop is perpendicular to the page. Then flux will be just *BA* = 5.0 × 10^{−5} T·π(0.20 m)^{2} = 6.3 × 10^{−6} T·m^{2} . (Be sure your units are right!)

(c) Induced EMF for this one loop is change in flux over time interval. It takes 1/500 of a second for the loop to make one complete rotation; so it takes ^{1} /_{4} of that, or 1/2000 of a second, for the loop to go from zero to maximum flux. Divide this change in flux by 1/2000 of a second … this is 6.3 × 10^{−6} T·m^{2} /0.0005 s = 0.013 V. (That”s 13 mV.)

(d) Now we can treat the circuit as if it were attached to a battery of voltage 13 mV. The equivalent resistance of the parallel combination of resistors *B* and *C* is 5 Ω; the total resistance of the circuit is 15 Ω. So the current in the whole circuit is 0.013 V/15 W = 8.4 × 10^{−4} A. (This can also be stated as 840 μA.) The current splits evenly between resistors *B* and *C* since they”re equal resistances, so we get 420 μA for resistor *C* .

** 5 .** The question might as well be restated, “Name four things you could do to change the flux through the loop,” because only a changing magnetic flux induces an EMF.

(a) Rotate the wire about an axis in the plane of the page. This will change the *θ* term in the expression for magnetic flux, *BA* cos *θ* .

(b) Pull the wire out of the field. This will change the area term, because the magnetic field lines will intersect a smaller area of the loop.

(c) Shrink or expand the loop. This also changes the area term in the equation for magnetic flux.

(d) Increase or decrease the strength of the magnetic field. This changes the *B* term in the flux equation.

** Rapid Review**

- Magnetic fields can be drawn as loops going from the north pole of a magnet to the south pole.
- A long, straight, current-carrying wire creates a magnetic field that wraps around the wire in concentric circles. The direction of the magnetic field is found by a right-hand rule.
- Similarly, loops of wire that carry current create magnetic fields. The direction of the magnetic field is, again, found by a right-hand rule.
- A magnetic field exerts a force on a charged particle if that particle is moving perpendicular to the magnetic field.
- When a charged particle moves perpendicular to a magnetic field, it ends up going in circles. This phenomenon is the basis behind mass spectrometry.
- A changing magnetic flux creates an induced EMF, which causes current to flow in a wire.
- Lenz”s Law says that when a changing magnetic flux induces a current, the direction of that current will be such that the magnetic field it induces is pointed in the opposite direction of the original change in magnetic flux.
- The Biot–Savart law has as its consequence that a little element of wire carrying a current produces a magnetic field that (1) wraps around the current element via the right-hand rule, and (2) decreases in magnitude as 1/
*r*^{2},*r*being the distance from the current element. This is applicable to Physics C only. - Ampére”s law has as its consequence that (1) the magnetic field produced by a very long, straight current is

outside the wire; inside the wire, the field increases linearly from zero at the wire”s center, and (2) the magnetic field produced by a wire-wrapped torus is zero everywhere outside the torus, but nonzero within the torus. The direction of the field inside the torus is around the donut.

^{1}^{ }If you”re not too impressed by these representations, just remember how physicists like to draw refrigerators. There”s a reason why these science folks weren”t accepted into art school.

^{2}^{ }You could also figure out the force on the left wire by using the same method we just used for the force on the right wire: draw the magnetic field produced by the right wire, and use the right-hand rule to find the direction of the magnetic force acting on the left wire.

^{3}^{ }But the calculus version of the induced EMF formula states: . If you”re given magnetic flux as a function of time, then take the *negative* time derivative to find the induced EMF.

^{4}^{ }“Yes.”

^{5}^{ }Which had first been accurately measured in the late 1600s using observations of the moons of Jupiter.

### CHAPTER 20

### Magnetism

** 1 .** Two long wires carry current perpendicular to the page in opposite directions as shown. The left wire has twice the current of the right wire. At which location will the magnetic field be closest to zero?

(A) A

(B) B

(C) C

(D) D

(E) E

** 2 .** A proton with a velocity (v) is moving directly away from a wire carrying a current (I) directed to the right in the +x direction, as shown in the figure. The proton will experience a force in which direction?

(A) –x

(B) –y

(C) –z

(D) +x

(E) +z

** 3 .** A parallel plate capacitor produces an electric field perpendicular to the magnetic field, as shown in the figure. The magnetic field is directed into the page in the –z direction. The magnetic and electric fields are adjusted so that a particle of charge +1

*e*, moving at a velocity of

*v*will pass straight through the fields in the +y direction. Which of the following changes will cause the particle to deflect to the left as it passes through the fields?

(A) Increasing the emf (ε) of the battery

(B) Doubling the charge to +2*e*

(C) Changing the sign of the charge to –1*e*

(D) Increasing the velocity of the particle

(E) Decreasing the magnetic field strength

** 4 .** A lab cart with a rectangular loop of metal wire fixed to the top travels along a frictionless horizontal track, as shown in the figure. While traveling to the right, the cart encounters a region of space with a strong magnetic field directed into the page. Which of the following graphs best depicts the velocity of the cart as a function of time as it enters, passes through, and finally exits the magnetic field on its journey?

(A)

(B)

(C)

(D)

(E)

** Answers**

** 1 .** E—Using the right hand rule (RHR) for magnetic fields around current carrying wires, we determine that the magnetic field rotates clockwise around the left wire and counterclockwise around the right wire. Thus, choices A, B, and C cannot be correct because the two B-fields combine in the downward direction. Thus, the viable options are choices D and E. Using the equation , we see that the left wire, with twice the current, must also have twice the radius to produce the same size B-field as the right wire: . Choice E satisfies this condition.

__2__ .**D** —Using the right hand rule (RHR) for a current carrying wire, we can determine that the magnetic field around the wire point is the –z direction in the vicinity of the proton. Using the RHR for forces on moving charges, we can determine that the proton will experience a magnetic force in the +x direction.

__3__ .**D** —For a charged particle to cross straight through crossed perpendicular magnetic and electric fields, the electric and magnetic forces on the charge must cancel each other out:

Note that since the velocity is perpendicular to the magnetic field, sinθ = sin(90°) = 1. Therefore, *E* = *vB* when the charge travels in a straight line thorough the fields. The electric force on the proton is directed to the right, while the magnetic force goes to the left. Therefore, to deflect the charge to the left, either the electric field must decrease, the magnetic field must increase, or the velocity of the charge must be increased. Note that changing the charge will have no effect because the charge *q* cancels out in the preceding equation.

__4__ .**B** —A retarding force will be present when there is a change in magnetic flux through the metal loop of wire:

This occurs only when the cart is entering the front edge and leaving the back edge of the field. No change in speed occurs while the cart is fully immersed in the magnetic field, as there is no change in flux. Therefore, choice A is incorrect because it shows no change in velocity. Choices D and E are also incorrect because they show only a single, continuous, retarding force slowing the cart down. Choice B is the best of the remaining options. The retarding force is

*F _{B} = ILB* sinθ =

*ILB*

and the current is given by

where the induced voltage is

ε = *V* = *BLv*

Combining these equations, we get the retarding force on the cart and loop:

*F _{B} *=

*B*

^{2}L^{2}vThis means the magnetic force and the acceleration of the cart are directly related to the velocity. The cart slows down when it enters the magnetic field and again when it leaves the field. Since the velocity of the cart when exiting the field will be less than when first entering the field, the retarding force will be less when exiting the field. The resulting acceleration, and thus the slope of the *v-t* graph, will be less when exiting the field.