Two identical conducting spheres are charged to +2Q and -Q respectively and are separated by a distance d (much greater than the radii of the spheres) as shown above. The magnitude of the force of attraction on the left sphere is F1. After the two spheres are made to touch and then are reseparated by distance d the magnitude of the force on the left sphere is F2. Which of the following relationships is correct? (A) 2F1 = F2 (B) F1 = F2 (C) F1 = 2F2 (D) F1 = 4F2 (E) F1 = 8F2

E) F1 = 8F2

Two small spheres have equal charges q and are separated by a distance d. The force exerted on each sphere by the other has magnitude F. If the charge on each sphere is doubled and d is halved the force on each sphere has magnitude (A) F (B) 2F (C) 4F (D) 8F (E) 16F

E) 16F

A charged particle traveling with a velocity v in an electric field E experiences a force F that must be (A) parallel to v (B) perpendicular to v (C) parallel to v x E (D) parallel to E (E) perpendicular to E

D) parallel to E

In which configuration of electric charges located at the vertices of an equilateral triangle is the electric field at P equal to zero? Point P is equidistant from the charges. (A) Configuration A (B) Configuration B (C) Configuration C (D) Configuration D (E) Configuration E

A) Configuration A

In which configuration of electric charges located at the vertices of an equilateral triangle is the electric field at P pointed at the midpoint between two of the charges? Point P is equidistant from the charges. (A) Configuration A (B) Configuration B (C) Configuration C (D) Configuration D (E) Configuration E

C) Configuration C

Positive charge Q is uniformly distributed over a thin ring of radius a that lies in a plane perpendicular to the x-axis with its center at the origin O as shown above. Which of the following graphs best represents the electric field along the positive x-axis? (A) Graph A (B) Graph B (C) Graph C (D) Graph D (E) Graph E

B) Graph B

From the electric field vector at a point one can determine which of the following? I. The direction of the electrostatic force on a test charge of known sign at that point II. The magnitude of the electrostatic force exerted per unit charge on a test charge at that point III. The electrostatic charge at that point (A) I only (B) III only (C) I and II only (D) II and III only (E) I II and III

C) I and II only

A circular ring made of an insulating material is cut in half. One half is given a charge -q uniformly distributed along its arc. The other half is given a charge +q also uniformly distributed along its arc. The two halves are then rejoined with insulation at the junctions J as shown above. If there is no change in the charge distributions what is the direction of the net electrostatic force on an electron located at the center of the circle? (A) Toward the top of the page (B) Toward the bottom of the page (C) To the right (D) To the left (E) Into the page

A) Toward the top of the page

A conducting sphere of radius R carries a charge Q. Another conducting sphere has a radius R/2 but carries the same charge. The spheres are far apart. The ratio of the electric field near the surface of the smaller sphere to the field near the surface of the larger sphere is most nearly (A) 1/4 (B) 1/2 (C) 1 (D) 2 (E) 4

E) 4

Two metal spheres that are initially uncharged are mounted on insulating stands as shown above. A negatively charged rubber rod is brought close to but does not make contact with sphere X. Sphere Y is then brought close to X on the side opposite to the rubber rod. Y is allowed to touch X and then is removed some distance away. The rubber rod is then moved far away from X and Y. What are the final charges on the spheres? (A) Sphere X Zero Sphere Y Zero (B) Sphere X Negative Sphere Y Negative (C) Sphere X Negative Sphere Y Positive (D) Sphere X Positive Sphere Y Negative (E) Sphere X Positive Sphere Y Positive

D) Sphere X Positive Sphere Y Negative

Two initially uncharged conductors 1 and 2 are mounted on insulating stands and are in contact as shown above. A negatively charged rod is brought near but does not touch them. With the rod held in place conductor 2 is moved to the right by pushing its stand so that the conductors are separated. Which of the following is now true of conductor 2? (A) It is uncharged (B) It is positively charged (C) It is negatively charged (D) It is charged but its sign cannot be predicted (E) It is at the same potential that it was before the charged rod was brought near

B) It is positively charged

As shown above two particles each of charge +Q are fixed at opposite corners of a square that lies in the plane of the page. A positive test charge +q is placed at a third corner. What is the direction of the force on the test charge due to the two other charges? (A) Direction A (B) Direction B (C) Direction C (D) Direction D (E) Direction E

E) Direction E

If F is the magnitude of the force on the test charge due to only one of the other charges what is the magnitude of the net force acting on the test charge due to both of these charges? (A) Zero (B) F/√2 (C) F (D) √2 F (E) 2F

D) √2 F

If the only force acting on an electron is due to a uniform electric field the electron moves with constant (A) acceleration in a direction opposite to that of the field (B) acceleration in the direction of the field (C) acceleration in a direction perpendicular to that of the field (D) speed in a direction opposite to that of the field (E) speed in the direction of the field

A) acceleration in a direction opposite to that of the field

Two charged particles each with a charge of +q are located along the x-axis at x = 2 and x = 4 as shown above. Which of the following shows the graph of the magnitude of the electric field along the x-axis from the origin to x = 6? (A) Graph A (B) Graph B (C) Graph C (D) Graph D (E) Graph E

A) Graph A

Particles of charge Q and -4Q are located on the x-axis as shown in the figure above. Which of the following describes the direction of the electric field at point P? (A) +x (B) +y (C) -y (D) Components in both the -x- and +y-directions (E) Components in both the +x- and -y-directions

E) Components in both the +x- and -y-directions

At which of the labeled points on the x-axis is the electric field zero? Charges are Q and -4Q. (A) A (B) B (C) C (D) D (E) E

A) A

When a negatively charged rod is brought near but does not touch the initially uncharged electroscope shown above the leaves spring apart (I). When the electroscope is then touched with a finger the leaves collapse (II). When next the finger and finally the rod are removed the leaves spring apart a second time (III). The charge on the leaves is (A) positive in both I and III (B) negative in both I and III (C) positive in I negative in III (D) negative in I positive in III (E) impossible to determine in either I or III

D) negative in I positive in III

Two infinite parallel sheets of charge perpendicular to the x-axis have equal and opposite charge densities as shown above. The sheet that intersects x = -a has uniform positive surface charge density; the sheet that intersects x = +a has uniform negative surface charge density. Which graph best represents the plot of electric field as a function of x? (A) Graph A (B) Graph B (C) Graph C (D) Graph D (E) Graph E

E) Graph E

A point charge is placed at the center of an uncharged spherical conducting shell of radius R. The electric fields inside and outside the sphere are measured. The point charge is then moved off center a distance R/2 and the fields are measured again. What is the effect on the electric fields? (A) Changed neither inside nor outside (B) Changed inside but not changed outside (C) Not changed inside but changed outside (D) Changed inside and outside (E) It cannot be determined without further information

B) Changed inside but not changed outside

The electric field E just outside the surface of a charged conductor is (A) directed perpendicular to the surface (B) directed parallel to the surface (C) independent of the surface charge density (D) zero (E) infinite

A) directed perpendicular to the surface

A closed surface in the shape of a cube of side a is oriented as shown above in a region where there is a constant electric field of magnitude E parallel to the x-axis. The total electric flux through the cubical surface is (A) -Ea² (B) zero (C) Ea² (D) 2Ea² (E) 6Ea²

B) zero

The figure above shows a spherical distribution of charge of radius R and constant charge density ρ. Which of the following graphs best represents the electric field strength E as a function of the distance r from the center of the sphere? (A) Graph A (B) Graph B (C) Graph C (D) Graph D (E) Graph E

A) Graph A

The electric field of two long coaxial cylinders is represented by lines of force as shown above. The charge on the inner cylinder is +Q. The charge on the outer cylinder is (A) +3Q (B) +Q (C) 0 (D) -Q (E) -3Q

E) -3Q

The net electric flux through a closed surface is (A) infinite only if there are no charges enclosed by the surface (B) infinite only if the net charge enclosed by the surface is zero (C) zero if only negative charges are enclosed by the surface (D) zero if only positive charges are enclosed by the surface (E) zero if the net charge enclosed by the surface is zero

E) zero if the net charge enclosed by the surface is zero

A solid nonconducting sphere of radius R has a charge Q uniformly distributed throughout its volume. A Gaussian surface of radius r with r < R is used to calculate the magnitude of the electric field E at a distance r from the center of the sphere. Which of the following equations results from a correct application of Gauss's law for this situation? (A) E(4πR²) = Q/εo (B) E(4πr²) = Q/εo (C) E(4πr²) = (Qr³)/(εoR³) (D) E(4πr²) = (Qr³)/(εoR³) (E) E(4πr²) = 0

C) E(4πr²) = (Qr³)/(εoR³)

The point charge Q shown above is at the center of a metal box that is isolated ungrounded and uncharged. Which of the following is true? (A) The net charge on the outside surface of the box is Q (B) The potential inside the box is zero (C) The electric field inside the box is constant (D) The electric field outside the box is zero everywhere (E) The electric field outside the box is the same as if only the point charge (and not the box) were there

A) The net charge on the outside surface of the box is Q

Gauss's law provides a convenient way to calculate the electric field outside and near each of the following isolated charged conductors EXCEPT a (A) large plate (B) sphere (C) cube (D) long solid rod (E) long hollow cylinder

C) cube

A point charge +Q is inside an uncharged conducting spherical shell that in turn is near several isolated point charges as shown above. The electric field at point P inside the shell depends on the magnitude of (A) Q only (B) the charge distribution on the sphere only (C) Q and the charge distribution on the sphere (D) all of the point charges (E) all of the point charges and the charge distribution on the sphere

A) Q only

A uniform spherical charge distribution has radius R. Which of the following is true of the electric field strength due to this charge distribution at a distance r from the center of the charge? (A) It is greatest when r = 0 (B) It is greatest when r = R/2 (C) It is directly proportional to r when r > R (D) It is directly proportional to r when r < R (E) It is directly proportional to r²

D) It is directly proportional to r when r < R

A distribution of charge is confined to a finite region of space. The difference in electric potential between any two points P1 and P2 due to this charge distribution depends only upon the (A) charges located at the points P1 and P2 (B) magnitude of a test charge moved from P1 to P2 (C) value of the electric field at P1 and P2 (D) path taken by a test charge moved from P1 to P2 (E) value of the integral -∫E·dr from P1 to P2

E) value of the integral -∫E·dr from P1 to P2

Two small spheres having charges of +2Q and -Q are 12 centimeters apart. The potential of points lying on a line joining the charges is best represented as a function of the distance x from the positive charge by which of the following? (A) Graph A (B) Graph B (C) Graph C (D) Graph D (E) Graph E

D) Graph D

In which configuration are both the electric field and the electric potential at the origin equal to zero? Five charge configurations on xy-plane using two or four point charges. (A) A (B) B (C) C (D) D (E) E

E) E

In which configuration is the value of the electric field at the origin equal to zero but the electric potential at the origin not equal to zero? (A) A (B) B (C) C (D) D (E) E

B) B

Two infinite parallel sheets of charge perpendicular to the x-axis have equal and opposite charge densities as shown above. Which graph best represents the plot of electric potential as a function of x? (A) Graph A (B) Graph B (C) Graph C (D) Graph D (E) Graph E

B) Graph B

An insulated spherical conductor of radius ro carries a charge q. The electric potential due to this system varies as a function of the distance r from the center of the sphere in which of the following ways? The potential is taken to be zero at r = ∞. (A) Graph A (B) Graph B (C) Graph C (D) Graph D (E) Graph E

A) Graph A

As shown in the diagram above a charged particle having mass m and charge -q is projected into the region between two parallel plates with a speed vo to the right. The potential difference between the plates is V and they are separated by a distance d. What is the net change in kinetic energy of the particle during the time it takes the particle to traverse the distance d? (A) +½mvo² (B) -qV/d (C) +qV/d (D) +qV (E) None of the above

D) +qV

Two conducting spheres one having twice the diameter of the other are shown above. The smaller sphere initially has a charge +q. When the spheres are connected by a thin wire which of the following is true? (A) 1 and 2 are both at the same potential (B) 2 has twice the potential of 1 (C) 2 has half the potential of 1 (D) 1 and 2 have equal charges (E) All of the charge is dissipated

A) 1 and 2 are both at the same potential

Points R and S are each the same distance d from two unequal charges +Q and +2Q as shown above. The work required to move a charge -Q from point R to point S is (A) dependent on the path taken from R to S (B) directly proportional to the distance between R and S (C) positive (D) zero (E) negative

D) zero

Two positive charges of magnitude q are each a distance d from the origin A of a coordinate system as shown above. At which of the following points is the electric potential greatest in magnitude? (A) A (B) B (C) C (D) D (E) E

A) A

Concentric conducting spheres of radii a and 2a bear equal but opposite charges +Q and -Q respectively. Which of the following graphs best represents the electric potential V as a function of r? (A) Graph A (B) Graph B (C) Graph C (D) Graph D (E) Graph E

D) Graph D

Which of the following statements about conductors under electrostatic conditions is true? (A) Positive work is required to move a positive charge over the surface of a conductor (B) Charge that is placed on the surface of a conductor always spreads evenly over the surface (C) The electric potential inside a conductor is always zero (D) The electric field at the surface of a conductor is tangent to the surface (E) The surface of a conductor is always an equipotential surface

E) The surface of a conductor is always an equipotential surface

A positive charge of 3.0 × 10⁻⁸ coulomb is placed in an upward directed uniform electric field of 4.0 × 10⁴ N/C. When the charge is moved 0.5 meter upward the work done by the electric force on the charge is (A) 6 × 10⁻⁴ J (B) 12 × 10⁻⁴ J (C) 2 × 10⁴ J (D) 8 × 10⁴ J (E) 12 × 10⁴ J

A) 6 × 10⁻⁴ J

Two conducting spheres X and Y have the same positive charge +Q but different radii (rx > ry) as shown above. The spheres are separated so that the distance between them is large compared with either radius. If a wire is connected between them in which direction will current be directed in the wire? (A) From X to Y (B) From Y to X (C) There will be no current in the wire (D) It cannot be determined without knowing the magnitude of Q (E) It cannot be determined without knowing whether the spheres are solid or hollow

B) From Y to X

A sphere of radius R has positive charge Q uniformly distributed on its surface. Which of the following represents the magnitude of the electric field E and the potential V as functions of r the distance from the center of the sphere when r < R? (A) E: 0 V: kQ/R (B) E: 0 V: kQ/r (C) E: 0 V: 0 (D) E: kQ/r² V: 0 (E) E: kQ/R² V: 0

A) E: 0 V: kQ/R

Which of the following represents the magnitude of the electric field E and the potential V as functions of r the distance from the center of sphere when r > R? (A) E: kQ/R² V: kQ/R (B) E: kQ/R V: kQ/R (C) E: kQ/R V: kQ/r (D) E: kQ/r² V: kQ/r (E) E: kQ/r² V: kQ/r²

D) E: kQ/r² V: kQ/r

Positive charge Q is uniformly distributed over a thin ring of radius a that lies in a plane perpendicular to the x-axis with its center at the origin O as shown above. The potential V at points on the x-axis is represented by which of the following functions? (A) V(x) = kQ/√(x²+a²) (B) V(x) = kQ/√(a²+x²) (C) V(x) = kQ/x² (D) V(x) = kQ/x (E) V(x) = kQ/(x+a)

B) V(x) = kQ/√(a²+x²)

Four positive charges of magnitude q are arranged at the corners of a square as shown above. At the center C of the square the potential due to one charge alone is Vo and the electric field due to one charge alone has magnitude Eo. Which of the following correctly gives the electric potential and the magnitude of the electric field at the center of the square due to all four charges? (A) Electric Potential: Zero Electric Field: Zero (B) Electric Potential: Zero Electric Field: 2Eo (C) Electric Potential: 2Vo Electric Field: 4Eo (D) Electric Potential: 4Vo Electric Field: Zero (E) Electric Potential: 4Vo Electric Field: 2Eo

D) Electric Potential: 4Vo Electric Field: Zero

Two charges -2Q and +Q are located on the x-axis as shown above. Point P at a distance of 3D from the origin O is one of two points on the positive x-axis at which the electric potential is zero. How far from the origin O is the other point? (A) (2/3)D (B) D (C) (3/2)D (D) (5/3)D (E) 2D

D) (5/3)D

What is the radial component of the electric field associated with the potential V = ar⁻² where a is a constant? (A) -2ar⁻³ (B) -2ar⁻¹ (C) ar⁻¹ (D) 2ar⁻¹ (E) 2ar⁻³

E) 2ar⁻³

Two concentric spherical conducting shells have radii r1 and r2 and charges Q1 and Q2 as shown above. Let r be the distance from the center of the spheres and consider the region r1 < r < r2. In this region the electric field is proportional to (A) Q1/r² (B) (Q1+Q2)/r² (C) (Q1+Q2)/r (D) Q1/r1+Q2/r (E) Q1/r+Q2/r²

A) Q1/r²

In this region the electric potential relative to infinity is proportional to (A) Q1/r² (B) (Q1+Q2)/r² (C) (Q1+Q2)/r (D) Q1/r1+Q2/r (E) Q1/r+Q2/r²

E) Q1/r+Q2/r²

A battery or batteries connected to two parallel plates produce the equipotential lines between the plates shown above. Which of the following configurations is most likely to produce these equipotential lines? (A) Config A (B) Config B (C) Config C (D) Config D (E) Config E

D) Config D

The force on an electron located on the 0-volt potential line is (A) 0 N (B) 1 N directed to the right (C) 1 N directed to the left (D) directed to the right but its magnitude cannot be determined without knowing the distance between the lines (E) directed to the left but its magnitude cannot be determined without knowing the distance between the lines

D) directed to the right but its magnitude cannot be determined without knowing the distance between the lines

The potential of an isolated conducting sphere of radius R is given as a function of the charge q on the sphere by the equation V = kq/R. If the sphere is initially uncharged the work W required to gradually increase the total charge on the sphere from zero to Q is given by which of the following expressions? (A) W = kQ/R (B) W = kQ²/R (C) W = ∫(kq/R)dq from 0 to Q (D) W = ∫(kq/R)dq from 0 to Q² (E) W = ∫(kq²/R)dq from 0 to Q

C) W = ∫(kq/R)dq from 0 to Q

Two charges located on a line where charge at point I is +3q and charge at point III is +2q. Point II is halfway between points I and III. Other than at infinity the electric field strength is zero at a point on the line in which of the following ranges? (A) To the left of I (B) Between I and II (C) Between II and III (D) To the right of III (E) None; the field is zero only at infinity

A) To the left of I

The electric potential is negative at some points on the line in which of the following ranges? (A) To the left of I (B) Between I and II (C) Between II and III (D) To the right of III (E) None; this potential is never negative

E) None; this potential is never negative

The graph above shows the electric potential V in a region of space as a function of position along the x-axis. At which point would a charged particle experience the force of greatest magnitude? (A) A (B) B (C) C (D) D (E) E

D) D

The work that must be done by an external agent to move a point charge of 2 mC from the origin to a point 3 m away is 5 J. What is the potential difference between the two points? (A) 4 × 10⁻⁴ V (B) 10⁻² V (C) 2.5 × 10³ V (D) 2 × 10⁶ V (E) 6 × 10⁶ V

C) 2.5 × 10³ V

Suppose that an electron (charge -e) could orbit a proton (charge +e) in a circular orbit of constant radius R. Assuming that the proton is stationary and only electrostatic forces act on the particles which of the following represents the kinetic energy of the two-particle system? (A) (1/4πεo)(e/R) (B) (1/8πεo)(e²/R) (C) -(1/8πεo)(e²/R) (D) (1/4πεo)(e²/R²) (E) -(1/4πεo)(e²/R²)

B) (1/8πεo)(e²/R)

In a region of space a spherically symmetric electric potential is given as a function of r the distance from the origin by the equation V(r) = kr² where k is a positive constant. What is the magnitude of the electric field at a point a distance ro from the origin? (A) Zero (B) kro (C) 2kro (D) kro² (E) 2kro³/3

C) 2kro

What is the direction of the electric field at a point a distance ro from the origin and the direction of the force on an electron placed at this point? (A) Electric Field: Toward origin Force on Electron: Toward origin (B) Electric Field: Toward origin Force on Electron: Away from origin (C) Electric Field: Away from origin Force on Electron: Toward origin (D) Electric Field: Away from origin Force on Electron: Away from origin (E) Electric Field: Undefined since the field is zero Force on Electron: Undefined since the force is zero

C) Electric Field: Away from origin Force on Electron: Toward origin

A positive electric charge is moved at a constant speed between two locations in an electric field with no work done by or against the field at any time during the motion. This situation can occur only if the (A) charge is moved in the direction of the field (B) charge is moved opposite to the direction of the field (C) charge is moved perpendicular to an equipotential line (D) charge is moved along an equipotential line (E) electric field is uniform

D) charge is moved along an equipotential line

The nonconducting hollow sphere of radius R shown above carries a large charge +Q which is uniformly distributed on its surface. There is a small hole in the sphere. A small charge +q is initially located at point P a distance r from the center of the sphere. If k = 1/4πεo what is the work that must be done by an external agent in moving the charge +q from P through the hole to the center O of the sphere? (A) Zero (B) kqQ/r (C) kqQ/R (D) kq(Q-q)/r (E) kqQ(1/R-1/r)

E) kqQ(1/R-1/r)

In a certain region the electric field along the x-axis is given by E = ax + b where a = 40 V/m² and b = 4 V/m. The potential difference between the origin and x = 0.5 m is (A) -36 V (B) -7 V (C) -3 V (D) 10 V (E) 16 V

B) -7 V

A 20 μF parallel-plate capacitor is fully charged to 30 V. The energy stored in the capacitor is most nearly (A) 9 × 10³ J (B) 9 × 10⁻³ J (C) 6 × 10⁻⁴ J (D) 2 × 10⁻⁴ J (E) 2 × 10⁻⁷ J

B) 9 × 10⁻³ J

A potential difference V is maintained between two large parallel conducting plates. An electron starts from rest on the surface of one plate and accelerates toward the other. Its speed as it reaches the second plate is proportional to (A) 1/V (B) 1/√V (C) √V (D) V (E) V²

C) √V

A solid metallic sphere of radius R has charge Q uniformly distributed on its outer surface. A graph of electric potential V as a function of position r is shown above. Which of the following graphs best represents the magnitude of the electric field E as a function of position r for this sphere? (A) Graph A (B) Graph B (C) Graph C (D) Graph D (E) Graph E

C) Graph C

As shown in the figure above six particles each with charge +Q are held fixed and are equally spaced around the circumference of a circle of radius R. What is the magnitude of the resultant electric field at the center of the circle? (A) 0 (B) 6Q/4πεoR² (C) 2√3Q/4πεoR² (D) 3√2Q/4πεoR² (E) 3Q/2πεoR²

A) 0

With the six particles held fixed how much work would be required to bring a seventh particle of charge +Q from very far away and place it at the center of the circle? (A) 0 (B) 6Q/4πεoR (C) 3Q²/2πεoR² (D) 3Q²/2πεoR (E) 9Q²/πεoR

B) 6Q/4πεoR

The diagram above shows equipotential lines produced by an unknown charge distribution. A B C D and E are points in the plane. Which vector below best describes the direction of the electric field at point A? (A) Vector A (B) Vector B (C) Vector C (D) Vector D (E) None of these; the field is zero

A) Vector A

At which point does the electric field have the greatest magnitude? (A) A (B) B (C) C (D) D (E) E

B) B

How much net work must be done by an external force to move a -1 μC point charge from rest at point C to rest at point E? (A) -20 μJ (B) -10 μJ (C) 10 μJ (D) 20 μJ (E) 30 μJ

B) -10 μJ

A physics problem starts: A solid sphere has charge distributed uniformly throughout. It may be correctly concluded that the (A) electric field is zero everywhere inside the sphere (B) electric field inside the sphere is the same as the electric field outside (C) electric potential on the surface of the sphere is not constant (D) electric potential in the center of the sphere is zero (E) sphere is not made of metal

E) sphere is not made of metal