MCAT Physics and Math Review

Chapter 5: Electrostatics and Magnetism

5.6 Magnetism

Any moving charge creates a magnetic field. Magnetic fields may be set up by the movement of individual charges, such as an electron moving through space; by the mass movement of charge in the form of a current though a conductive material, such as a copper wire; or by permanent magnets. The SI unit for magnetic field strength is the tesla (T), where  The size of the tesla unit is quite large, so small magnetic fields are sometimes measured in gauss, where 1 T = 104 gauss.

KEY CONCEPT

Any moving charge, whether a single electron traveling through space or a current through a conductive material, creates a magnetic field. The SI unit for magnetic field strength is the tesla (T).

All materials can be classified as diamagnetic, paramagnetic, or ferromagnetic. Diamagnetic materials are made of atoms with no unpaired electrons and that have no net magnetic field. These materials are slightly repelled by a magnet and so can be called weakly antimagnetic. Diamagnetic materials include common materials that you wouldn’t expect to get stuck to a magnet: wood, plastics, water, glass, and skin, just to name a few.

The atoms of both paramagnetic and ferromagnetic materials have unpaired electrons, so these atoms do have a net magnetic dipole moment, but the atoms in these materials are usually randomly oriented so that the material itself creates no net magnetic field. Paramagnetic materials will become weakly magnetized in the presence of an external magnetic field, aligning the magnetic dipoles of the with the external field. Upon removal of the external field, the thermal energy of the individual atoms will cause the individual magnetic dipoles to reorient randomly. Some paramagnetic materials include aluminum, copper, and gold.

Ferromagnetic materials, like paramagnetic materials, have unpaired electrons and permanent atomic magnetic dipoles that are normally oriented randomly so that the material has no net magnetic dipole. However, unlike paramagnetic materials, ferromagnetic materials will become strongly magnetized when exposed to a magnetic field or under certain temperatures. Common ferromagnetic materials are include iron, nickel and cobalt. Bar magnets are ferromagnetic materials with a north and south pole. Field lines exit the north pole and enter the south pole. Because magnetic field lines are circular, it is impossible to have a monopole magnet. If two bar magnets are allowed to interact, opposite poles will attract each other, while like poles will repel each other.

MAGNETIC FIELDS

Because any moving charge creates a magnetic field, we would certainly expect that a collection of moving charge, in the form of a current through a conductive wire, would produce a magnetic field in its vicinity. The configuration of the magnetic field lines surrounding a current-carrying wire will depend on the shape of the wire. Two special cases that are commonly tested on the MCAT include a long, straight wire and a circular loop of wire (with particular attention paid to the magnetic field at the center of that loop).

For an infinitely long and straight current-carrying wire, we can calculate the magnitude of the magnetic field produced by the current I in the wire at a perpendicular distance, r, from the wire as:

Equation 5.11

where B is the magnetic field at a distance r from the wire, and µ0 is the permeability of free space (  and I is the current. The equation demonstrates an inverse relationship between the magnitude of the magnetic field and the distance from the current. Straight wires create magnetic fields in the shape of concentric rings. To determine the direction of the field vectors, use a right-hand rule. (This is one of two right-hand rules used in magnetism.) Point your thumb in the direction of the current and wrap your fingers around the current-carrying wire. Your fingers then mimic the circular field lines, curling around the wire.

For a circular loop of current-carrying wire of radius r, the magnitude of the magnetic field at the center of the circular loop is given as:

Equation 5.12

You’ll notice that the two equations are quite similar—the obvious difference being that the equation for the magnetic field at the center of the circular loop of wire does not include the constant π. The less obvious difference is that the first expression gives the magnitude of the magnetic field at any perpendicular distance, r, from the current-carrying wire, while the second expression gives the magnitude of the magnetic field only at the center of the circular loop of current-carrying wire with radius r.

Example:

Suppose a wire is formed into a loop that carries a current of 0.25 A in a clockwise direction, as shown here:

Determine the direction of the magnetic field produced by this loop within the loop and outside the loop. If the loop has a diameter of 1 m, what is the magnitude of the magnetic field at the center of the loop?

Solution:

Use the right-hand rule to determine the direction of the magnetic field within and outside the loop, as shown here:

Align your right thumb with the current at any point in the loop. When you encircle the wire with the remaining fingers of your right hand, your fingers should point into the page within the loop and out of the page outside of the loop.

To determine the magnitude of the magnetic field at the center, use the equation for a loop of wire?

MAGNETIC FORCES

Now that we’ve reviewed the ways in which magnetic fields can be created, let’s examine the forces that are exerted by magnetic fields on moving charges. Magnetic fields exert forces only on other moving charges. That is, charges do not “sense” their own fields; they only sense the field established by some external charge or collection of charges. Therefore, in our discussion of the magnetic force on moving charges and on current-carrying wires, we will assume the presence of a fixed and uniform external magnetic field. Note that charges often have both electrostatic and magnetic forces acting on them at the same time; the sum of these electrostatic and magnetic forces is known as the Lorentz force.

Force on a Moving Charge

When a charge moves in a magnetic field, a magnetic force may be exerted on it, the magnitude of which can be calculated as follows:

FB = qvB sinθ

Equation 5.13

where q is the charge, ν is the magnitude of its velocity, B is the magnitude of the magnetic field, and θ is the smallest angle between the vector v and the magnetic field vector B. Notice that the magnetic force is a function of the sine of the angle, which means that the charge must have a perpendicular component of velocity in order to experience a magnetic force. If the charge is moving parallel or antiparallel to the magnetic field vector, it will experience no magnetic force.

KEY CONCEPT

Remember that sin 0° and sin 180° equal zero. This means that any charge moving parallel or antiparallel to the direction of the magnetic field will experience no force from the magnetic field.

Here we will introduce the second right-hand rule that you should practice in anticipation of Test Day. To determine the direction of the magnetic force on a moving charge, first position your right thumb in the direction of the velocity vector. Then, put your fingers in the direction of the magnetic field lines. Your palm will point in the direction of the force vector for a positive charge, whereas the back of your hand will point in the direction of the force vector for a negative charge.

MNEMONIC

Parts of the right-hand rule for magnetic force:

·        Thumb—velocity (indicates direction of movement, like a hitchhiker’s thumb)

·        Fingers—field lines (fingers are parallel like the uniform magnetic field lines)

·        Palm—force on a positive charge (you might give a “high five” to a positive person)

·        Back of hand—force on a negative charge (you might give a backhand to a negative person)

Example:

Suppose a proton is moving with a speed of  toward the top of the page through a uniform magnetic field of 3.0 T directed into the page, as shown here:

What is the magnitude and direction of the magnetic force on the proton? Describe the motion that will result from this setup. (Note: The charge of a proton is 1.60 × 10−19 C, and its mass is 1.67 × 10−27 kg.)

Solution:

Start by determining the magnitude of the force:

To determine the direction, use the right-hand rule. Your thumb should point up the page in the direction of v. Your fingers should point into the page in the direction of B. Protons are positively charged; thus the force, FB, is in the direction of your palm, which is to the left. Note thatv and FB will always be perpendicular to each other; this implies that uniform circular motion will occur in this field, with FB pointing radially inward toward the center of the circle.

If the centripetal force is the magnetic force, then we can set these two equations equal to each other:

Thus, the proton will move in a circle with a radius of 52 nm.

Force on a Current-Carrying Wire

We’ve just examined the force that can be created by a magnetic field when a point charge moves through the field, so it should not come as a surprise that a current-carrying wire placed in a magnetic field may also experience a magnetic force. For a straight wire, the magnitude of the force created by an external magnetic field, FB, is:

FB = ILB sinθ

Equation 5.14

where I is the current, L is the length of the wire in the field, B is the magnitude of the magnetic field, and θ is the angle between L and B. The same right-hand rule can be used for a current-carrying wire in a field as for a moving point charge; just remember that current is considered the flow of positive charge.

Example:

Suppose a wire of length 2.0 m is conducting a current of 5.0 A toward the top of the page and through a 30 gauss uniform magnetic field directed into the page. What is the magnitude and direction of the magnetic force on the wire?

Solution:

Because 1 T = 104 gauss, 1 gauss = 10−4 T, and 30 gauss = 30 × 10−4 T = 3 × 10−3 T. The wire is conducting current toward the top of the page, and the magnetic field points into the page; therefore, the current is perpendicular to the magnetic field. The angle between them is θ = 90°. Now, plug into the equation:

To determine the direction, use the right-hand rule. Your thumb should point up the page in the direction of L. Your fingers should point into the page in the direction of B. Current is a flow of positive charge; thus, the force, FB, is in the direction of your palm, which is to the left.

MCAT Concept Check 5.6:

Before you move on, assess your understanding of the material with these questions.

1.    What are the requirements to have a nonzero electric field? A nonzero magnetic field? A nonzero magnetic force?

·        Nonzero electric field:

·        Nonzero magnetic field:

·        Nonzero magnetic force:

2.    Which would experience a larger magnetic field: an object placed five meters to the left of a current carrying wire, or an object placed at the center of a circle with a radius of five meters. (Note: Assume the current is constant;

3.    For each of the following combinations of velocity and magnetic field directions, determine the direction of the magnetic force on the given particle:

v

B

Particle

F

Up the page

Left

Electron

 

Into the page

Out of the page

Proton

 

Right

Into the page

Proton

 

Out of the page

Left

Electron

 

Down the page

Right

Neutron