MCAT Physics and Math Review

Chapter 5: Electrostatics and Magnetism

Conclusion

In this chapter, we reviewed the very notion of charge, reminding ourselves that charge comes in two varieties: positive and negative. We also explored the fact that charges travel differently within insulators and conductors. We learned that charges establish electric fields through which they can exert forces on other charges. We relied on similarities between electrical and gravitational systems to better understand Coulomb’s law and the nature of the forces that exist between charged particles. Don’t forget that electrical forces can be repulsive as well as attractive, which is one of the differences between electrical and gravitational systems. Charges contain electrical potential energy, which we defined as their energy of position with respect to other charges. Charges move within an electric field from one position of electrical potential to another; they will move spontaneously through an electrical potential difference, or voltage, in whichever direction results in a decrease in the charge’s electrical potential energy. Then, we considered the geometry of the electric dipole and derived the equation for calculating the electrical potential at any point in space around the dipole. Finally, we considered magnetic fields and forces. In the next chapter, we’ll examine moving charges as they interact with circuit elements and complete our understanding of electricity.

Concept Summary

Charges

·        The SI unit of charge is the coulomb.

·        Protons have a positive charge and electrons have a negative charge.

o   Both protons and electrons possess the fundamental unit of charge (e = 1.60 × 10–19 C).

o   Protons and electrons have different masses.

·        Opposite charges exert attractive forces, and like charges exert repulsive forces.

·        Conductors allow the free and uniform passage of electrons when charged.

·        Insulators resist the movement of charge and will have localized areas of charge that do not distribute over the surface of the material.

Coulomb’s Law

·        Coulomb’s law gives the magnitude of the electrostatic force vector between two charges. The force vector always points along the line connecting the centers of the two charges.

·        Every stationary charge generates an electric field, which can exert forces on other charges.

·        The electric field is the ratio of the force that is exerted on a test charge to the magnitude of that charge.

o   Electric field vectors can be represented as field lines that radiate outward from positive source charges and radiate inward to negative source charges.

o   Positive test charges will move in the direction of the field lines; negative test charges will move in the direction opposite of the field lines.

Electrical Potential Energy

·        Electrical potential energy is the amount of work required to bring the test charge from infinitely far away to a given position in the vicinity of a source charge.

·        The electrical potential energy of a system will increase when two like charges move toward each other or when two opposite charges move further apart.

·        The electrical potential energy of a system will decrease when two opposite charges move toward each other or when two like charges move further apart.

Electrical Potential

·        Electrical potential is the electrical potential energy per unit charge.

·        Different points in the space of an electric field surrounding a source charge will have different electrical potential values.

·        Potential difference (voltage) is the change in electrical potential that accompanies the movement of a test charge from one position to another.

o   Potential difference is path independent and depends only on the initial and final positions of the test charge.

o   The units for both electrical potential and voltage are volts.

·        Test charges will move spontaneously in whichever direction results in a decrease in their electrical potential energy.

o   Positive test charges will move spontaneously from high potential to low potential.

o   Negative test charges will move spontaneously from low potential to high potential.

Special Cases in Electrostatics

·        Equipotential lines designate the set of points around a source charge or multiple source charges that have the same electrical potential.

o   Equipotential lines are always perpendicular to electric field lines.

o   Work will be done when a charge is moved from one equipotential line to another; the work is independent of the pathway taken between the lines.

o   No work is done when a charge moves from a point on an equipotential line to another point on the same equipotential line.

·        Two charges of opposite sign separated by a distance d generate an electric dipole.

o   In an external electric field, an electric dipole will experience a net torque until it is aligned with the electric field vector.

o   An electric field will not induce any translational motion in the dipole regardless of its orientation with respect to the electric field vector.

Magnetism

·        Magnetic fields are created by magnets and moving charges.

·        The SI unit for the magnetic field is the tesla (T; 1 T = 1000 gauss).

·        Diamagnetic materials possess no unpaired electrons and are slightly repelled by a magnet.

·        Paramagnetic materials possess some unpaired electrons and become weakly magnetic in an external magnetic field.

·        Ferromagnetic materials possess some unpaired electrons and become strongly magnetic in an external magnetic field.

·        Magnets have a north and a south pole; field lines point from the north to the south pole.

·        Current-carrying wires create magnetic fields that are concentric circles surrounding the wire.

·        External magnetic fields exert forces on charges moving in any direction except parallel or antiparallel to the field.

·        Point charges may undergo uniform circular motion in a uniform magnetic field wherein the centripetal force is the magnetic force acting on the point charge.

·        The direction of the magnetic force on a moving charge or current-carrying wire is determined using the right-hand rule.

·        The Lorentz force is the sum of the electrostatic and magnetic forces acting on a body.

Answers to Concept Checks

·        5.1

1.    The electrons will experience the greater acceleration because they are subject to the same force as the protons but have a significantly smaller mass.

2.    Conductors: blood, copper, iron, sulfuric acid; insulators: hair, glass, distilled water

3.    The net charge will be –1 C; neutrons do not contribute charge.

·        5.2

1.    The electric field would be 0 because the two charges are the same. In this case, the fields exerted by each charge at the midpoint will cancel out and there will be no electric field.

2.    For a pair of charges, a negative electrostatic force points from one charge to the other (attractive), while a positive electrostatic force points from one charge away from the other (repulsive).

3.   

4.    Electrostatic force is directly related to each charge and related to the distance by an inverse square relationship. Electric field is unrelated to test charge but is still related to distance by an inverse square relationship. Note that it is the source charge that creates the electric field—not the test charge—so we cannot use the equation  to determine a relationship.

·        5.3

1.    A decrease in potential energy indicates that the system has become more stable. Keep in mind that negative numbers with larger absolute values are more negative, and represent a decrease in value from negative numbers with smaller absolute values (that is, –4 > −7 even though |−4| < |−7|).

2.    Electrical potential energy is Coulomb’s multiplied by distance, whereas gravitational potential energy is the universal law of gravitation multiplied by distance.

3.    If both particles have the same charge, the electrical potential energy decreases as distance increases. If the two particles have opposite charges, then the electrical potential energy increases as distance increases.

·        5.4

1.    Electrical potential is the ratio of a charge’s electrical potential energy to the magnitude of the charge itself. Voltage, or potential difference, is a measure of the change in electrical potential between two points, which provides an indication of the tendency toward movement in one direction or the other.

2.    A charge will move in such a way to minimize its potential energy. Placing a charge at a point of zero electrical potential does not indicate that there is zero potential difference, so the charge may or may not move—and if it moves, it may move toward or away from the source charge depending on the sign of the source charge and test charge.

3.    True. Electrical potential energy is measured in joules (J), while electrical potential and potential difference (voltage) are measured in volts (V).

·        5.5

1.    Equipotential lines are the sets of points within space at which the potential difference between any two points is zero. This is best visualized as concentric spheres surrounding a source charge. An electric dipole is the separation of charge within a molecule such that there is a permanent or temporary region of equal and opposite charges at a particular distance.

2.    There is no voltage between two points on an equipotential line, so there will be no acceleration along the line. However, there is a potential difference between different sets of equipotential lines, which can cause particles to move and accelerate.

3.    The perpendicular bisector of an electric dipole is an equipotential plane that is perpendicular to the axis of the dipole. As such, the equation  is necessarily equal to 0 because cos 90° = 0.

4.    A dipole will rotate within an external electric field such that its dipole moment aligns with the field.

·        5.6

1.    To create an electric field, one needs a charge. To create a magnetic field, one needs a charge that must also be moving. To create a magnetic force, one needs an external electric field acting on a charge moving any direction except parallel or antiparallel to the external field.

2.    We need not determine the actual values of the magnetic fields in these two cases and can compare the two equations instead. The magnetic field created by the current-carrying wire is given by  the magnetic field created by the loop of wire is given by  μ0,I, and r are the same in both equations. Therefore, the magnetic field at the center of the loop must be larger because the denominator in that equation does not include π.

3.     

v

B

Particle

F

Up the page

Left

Electron

Into the page

Into the page

Out of the page

Proton

None (sin 180° = 0)

Right

Into the page

Proton

Up the page

Out of the page

Left

Electron

Up the page

Down the page

Right

Neutron

None (q = 0)

Equations to Remember

(5.1) Coulomb’s law

(5.2) Electric field

(5.3) Electrical potential energy

(5.4) Electrical potential (from electrical potential energy)

(5.5) Electrical potential (from source charge)

(5.6) Voltage

(5.7) Electrical potential near a dipole

(5.8) Dipole momentp = qd

(5.9) Electric field on the perpendicular bisector of a dipole

(5.10) Torque on a dipole in an electric fieldτ = pE sin θ

(5.11) Magnetic field from a straight wire

(5.12) Magnetic field from a loop of wire

(5.13) Magnetic force on a moving point chargeFB = qvB sinθ

(5.14) Magnetic force on a current-carrying wireFB = ILB sinθ

Shared Concepts

·        General Chemistry Chapter 1

o   Atomic Structure

·        General Chemistry Chapter 3

o   Bonding and Chemical Interactions

·        General Chemistry Chapter 12

o   Electrochemistry

·        Physics and Math Chapter 1

o   Kinematics and Dynamics

·        Physics and Math Chapter 2

o   Work and Energy

·        Physics and Math Chapter 6

o   Circuits