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Physics 106

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 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?


(a) What is the potential energy of a system of a charge -q at a distance r from a charge +Q? (b) How much work is required to move a charge -q around a circular path which has +Q at the center of the circle?


A proton of charge +e gains an energy of 4.8 x 10-19 J when it is accelerated through a potential difference Vab.  (a) What is Vab?  (b) An alpha particle of charge +2e is accelerated through the same potential difference. How much energy does it acquire?


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.


A constant electric field E = 104 N/C exists between two parallel plates which are separated by 0.10 m. (a) How much work is done to carry a charge of
+10-6 C from the negative plate to the positive plate without increasing its kinetic energy? (b) What is the potential energy difference between the plates? (c) What is the potential difference between the plates?


A particle of charge q = 10-6 C is placed in an electric field E = 105 N/C directed at an angle of 53o below the horizontal (Fig. 5 below).  (a) How much work is done to carry the charge from B to A in Fig. 5 without increasing its kinetic energy? (b) What is the potential energy difference UA - UB?  (c) What is the potential difference VAB?


In Fig. 6 below, q1 = -4 µC and q2 = +2 µC. Find (a) the potential at B, (b) the potential at A,  (c) the work done to move a particle with q3 = +3 µC at a constant speed from B to A. (d) Does the work done depend on the path from B to A?


Assume that an electron of mass m moves in a circle of radius r with a constant speed around a stationary proton (Fig. 7a 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. 7b above, the electron's motion can be approximated by motion back and forth along a straight line (Fig. 7c above). 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 13.


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 of (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 κ = 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 κ, as shown in the Fig. 8 below.


Find the equivalent capacitance of the circuit in Fig. 9 below and the charge on each capacitor.


Find (a) the equivalent capacitance of the combination of capacitors shown in Fig. 10 below,  (b) the charge on each capacitor, and (c) the potential difference VAD and VDB. (d) How much energy is stored in the capacitors?

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Susan D. Kunk
Phyllis J. Fleming
August 8, 2002
December 27, 2002