1. Equipotential contours for three point charges (red/dark circles) are shown in the figure, along with the trajectory (red dotted line) of an ion with a charge of -2e that was passing through the region. (10 points) (a) Sketch the electric field lines of the three point charges, including arrows to indicate its direction. Your field lines needs to cover the region so: • there should be field lines on ALL sides of EACH charge • field lines should extend past the outermost contours. +30 V +15 V +10 V +5 V- 0 V- +15 V -15 V -5 V -O V +30 V (b) How much did the electric potential energy stored in the electric field of the three charges change as the ion moved from point a to point b along the (red dotted) path shown in the figure? Give your answer in eV.

When charged particles are separated by an infinite distance, the electric potential energy of the pair is zero. When the particles are brought close, the electric potential energy of a pair with the same sign is positive, whereas the electric potential energy of a pair with opposite signs is negative. Give a physical explanation of this statement.

Two charges 20Cand+2.0C are separated by 4.0 cm on the z-axis symmetrically about origin, with the positive one uppermost. Two space points of interest and P1andP2 are located 3.0 cm and 30 cm from origin at an angle 30 with respect to the z-axis. Evaluate electric potentials at P1andP2 and in two ways: (a) Using the exact formula for point charges, and (b) rising the approximate dipole potential formula.

It the voltage between two points is zero, can a test charge be moved between them with zero net work being done? Can this necessarily be done without exerting a force? Explain.

An electron is released from rest in a uniform electric field. Determine whether the following quantities increase, decrease, or remain unchanged as the electron moves. Indicate your answers with I (increase), D (decrease), or U (unchanged), respectively. (a) The electric potential at the electrons location (b) The electrons associated electric potential energy (c) Its kinetic energy (d) Its total energy.

Consider two conducting spheres with radii R1 and R2 separated by a distance much greater than cither radius. A total charge Q is shared between the spheres. We wish to show that when the electric potential energy of the system has a minimum value, the potential difference between the spheres is zero. The total charge Q is equal to q1 + q2, where q1 represents the charge on the first sphere and q2 the charge on the second. Because the spheres are very far apart, you can assume the charge of each is uniformly distributed over its surface. (a) Show that the energy associated with a single conducting sphere of radius R and charge q surrounded by a vacuum is UE = keq2/2R. (b) Find the total energy of the system of two spheres in terms of the total charge Q, and the radii and R1 and R2. (c) To minimize the energy, differentiate the result to part (b) with respect to q1 and set the derivative equal to zero. Solve for q1 in terms of Q and the radii. (d) From the result to part (c), find the charge q2. (e) Find the potential of each sphere. (f) What is the potential difference between the spheres?

Two very large metal plates are placed 2.0 cm apart, with a potential difference of 12 V between them. Consider one plate to be at 12 V, and the other at 0 V. (a) Sketch the equipotential surfaces for 0, 4, 8, and 12 V. (b) Next sketch in some electric field lines, and confirm that they are perpendicular to the equipotential lines.

Explain in your own words why equipotential lines and surfaces must be perpendicular to electric field lines.

If the voltage between two points is zero can a test charge be moved between them with zero net work being done? Can this necessarily be done without exerting a force? Explain.

Can the potential of a non-uniformly charged sphere be the same as that of a point charge? Explain.