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Physics 108

Review - Potential Energy, Potential, and Capacitors


Determine the electric potential due to two point charges,  Q1 and Q2,  along a perpendicular bisector at point P of the line joining the charges (Fig. 1 below). (b) What is the potential at P when r >> a?


Find (a) the electric potential at point P in Fig. 2 below. (b) Find the work done in bringing up a charge of +3 nC from infinity. (c) Repeat (a) and (b) for
q2 = +1 nC.


In Fig. 3 below, q1 = q2 = -200µC, q3 = q4 = +100mC and the charge at the center of the square q5 = +20µC.  With q5 removed in Fig. 3, find (a) the potential at the center of the square and (b) the work done to bring q5 from infinity to the center of the square.


A positively charged particle with q = +10-6 C and mass m = 10-3 kg is placed between the plates of a parallel plate capacitor and is at rest. (a) Draw the capacitor with charges, label the forces acting on the particle, and calculate the electric field between the plates. (b) Given the distance between the plates as
10-3 m,  find the potential difference across the plates.


An alpha particle (charge +2e) approaches a gold nucleus (charge +79e) from a very great distance, starting with kinetic energy K. The alpha particle just touches the surface of the nucleus (the radius of the gold nucleus is
7.0 x 10-15 m) where its velocity is reversed. Find the initial kinetic energy of the alpha particle.


A proton of mass m = 1.67 x 10-27 kg and charge q = e = 1.60 x 10-19 C is accelerated from rest through a potential difference of 100 V.  What velocity does it achieve?


Under undisturbed conditions there is a downward-directed electric field above the surface of the earth.  If y represents the height in meters measured from the surface, Ey, the magnitude of the field over a limited range in y, can be represented as Ey = - (300 V/m - 0.010y V/m2). Over the range of validity of Ey, obtain an expression for the potential above the surface, taking the surface to be zero potential.


In the figure below, the nonconducting disk of radius R has a uniform charge density σ on one surface. Find (a) the area dA of the ring of radius r and thickness dr, (b) the charge element dq on the ring of radius r, (c) the potential due to the ring and the potential due to the disk at point P in the figure. (d) Find Ez from -  ΔV/ Δz.


A point charge q1 = +80 nC is situated on the X-axis at the origin and a second point charge q2 = - 60 nC is placed at x = +0.20 m, as shown in the figure below. The field point A is located on the x-axis at x = 0.10 m.  A second field point B is in the X-Y plane at a distance of 0.16 m from q1 and 0.12 m from q2. Find (a) the potential at point A, (b) the potential at point B, and (c) the work done in transferring a charge of +10 µC from B to A.


Two long, thin charged wires intersect the plane of the paper normally at A and B in the figure below. The charge per unit length on each wire is λ. Find the potential difference between the points C and D.  CD is the perpendicular bisector of AB.  Let AC = CB = 4.00 m, CD = 10.12 m, and AD = 10.88 m.


Each of three parallel plate capacitors has area A and spacing d.  Find the spacing d’ of a single capacitor of plate area A if its capacitance equals that of the three connected in (a) parallel (b) series.


A spherical capacitor of radius R1 is charged to a potential difference (Vab)i.
The charging battery is then disconnected and the capacitor is connected in parallel with a second (initially uncharged) spherical capacitor of radius R2. The measured potential difference drops to (Vab)f.  Find R2 in terms of R1, (Vab)i and (Vab)f.


A parallel plate capacitor has a capacitance C when the plates have an area A, a plate separation d, and the plates are in a vacuum. The charge on the plates is Q when a battery of potential difference of Vab  is placed and kept across the capacitor. Find what happens to (i) the capacitance, (ii) charge and (iii) the electric field when (only one at a time), (a) the plate separation is doubled and (b) a dielectric of constant k= 2 is inserted between the plates.


Determine the capacitance of a parallel plate capacitor in which the region between the plates is partially filled with a dielectric slab of thickness t and dielectric constant k, as shown in the figure below.


Find (a) the equivalent capacitance of the combination of capacitors shown in Fig. 4, (b) the charge on each capacitor, and (c) the potential difference VAD and VDB. (d) How much energy is stored in the capacitors?


Find the electric potential for a uniform spherical charge distribution of radius R for  (a) r ≥ R,  (b) r ≤ R.  (c) Sketch graphs of E and  V vs r.  How can you find the electric field E from a knowledge of V(r)?


A cylindrical capacitor consists of an inner conducting cylinder of radius a with charge +Q and an outer conducting cylinder of radius b with charge -Q. The length L of the cylinders is great enough that the electric field between the two cylinders is radial.  Find C for this capacitor.


Parallel plates carry charges ±Q and have a separation s. To move the plates an additional distance ds, takes dW = F ds. (a) What is the energy U of the plates with charge q, area A, and distance of separation s? The work done in the separation of the plates would appear as an increase dU in the potential energy. (b) Write an expression for the increase in potential energy. (c) Combine all of this to find the force required to maintain a separation s between the plates.


A spherical volume of radius R is filled with a uniform charge density ρ. To find the potential energy of this sphere, we find the work to assemble it.  Find (a) the charge q in a sphere of radius r, (b) the charge q’ in a spherical shell of radius r and thickness dr, (c) the work done U in bringing an infinitesimal shell of thickness dr from infinity to the radius r, (d) integrate the expression in (c) from r = 0  to  r = R  to find U and express it in terms of the total charge Q in the sphere of radius R.


Around 1900 a number of physicists believed that the rest mass of the electron might be purely electrical in origin. Using the result of Problem 19,  set the potential energy of the spherically charged volume equal to mc2 and see what you find for the radius ro of the electron. Obviously there is a problem with the model since no mechanism is provided to hold the charge together.


A negatively charged particle of charge q = -10-6 C and mass m = 10-10 kg travels in an elliptical orbit around a positive charge q’ = +10-6 C.  When it is at A, as shown in Fig. 5 below, the distance from -q to +q is 6 x 10-5 m. When it is at B, the distance from -q to +q is 3 x 10-5 m. (a) Use conservation of angular momentum to find the ratio of the particle's velocity at B to its velocity at A. (b) Use conservation of energy to find va and vb.


Assume that an electron of mass m moves in a circle of radius r with a constant speed around a stationary proton (Fig. 6a below). (a) What is the electric potential energy of this system in terms of k, e, and r? (b) The electric force of the proton on the electron provides the centripetal acceleration. Use Newton's second law of motion to find mv2.  Find (c) the total energy and the angular momentum of the system.


In one model of the hydrogen atom, the electron moves in an elliptical orbit around a stationary proton. For the very "skinny" ellipse of Fig. 6b below, the electron's motion can be approximated by motion back and forth along a straight line (Fig. 6c below).  Find (a) the electric potential energy of the system when the electron is a distance 2r from the proton and (b) the kinetic energy of the system when the electron is at this distance. (c) Compare the total energy and the angular momentum of the system with the model of Problem 22.


An insulating washer (Fig. 7) carries charge Q uniformly distributed along its surface. Find (a) the potential V at point P a height y above the surface, (b) the electric field at point P (magnitude and direction) and (c) the electric field at O.


A conducting sphere with radius R carrying a charge qR is surrounded by a spherical shell with inner radius a and outer radius b, as illustrated in Fig. 8 below. The outer surface of the shell has a charge qb = 6 µC and its inner surface has charge qa = 5 µC.

(a) Find q and explain how you found it.
(b) Find E as a function of r for  (i) 0 < r < R,  (ii) R < r < a,
     (iii) a < r < b, and (iv) r > b.

(c) Sketch a graph of (i) E versus r,  and (ii) V versus r.


In Figure 9, an infinite line carries a charge per unit length λ. The point P is distance a from the infinite line and is grounded so that  V(a) = 0.

(a) Find V(r) for r < a.  (b) Explain why you cannot take a = infinity, and (c) verify that Er = -dV/dr.


Figure 10 is a graph of electric field as a function of x. Plot V as a function of x. Ignore the points on the graph where there are discontinuities.

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Susan D. Kunk
Phyllis J. Fleming
August 8, 2002
February 14, 2003