Download Problem Set 4

Problem Set 4
Due: see website for due dates
Chapter 24: Electric Potential
Exercises & Problems: 9, 19, 57, 59, 66, 67, 95
Question A
(i) In electrostatics, why is a conductor an equipotential surface? What if the
conductor is in a nonequilibrium situation (an external E-field is present), will the
conductor still be an equipotential surface? (It is not sufficient to say because
Einside = 0). (ii) What does it mean that it takes zero work to move a charge
across a conductor?
Question B
(i) Assume that the binding energy is equivalent to the electrical potential energy for an
atom. If the nitrogen gas molecule (N2) has a separation distance of 0.11 nm between
the nucleus and an outer electron, what is the binding energy of this electron? Draw a
picture of this system. (ii) Now explain how the N2 gas molecules are ionized when
dielectric breakdown occurs using the electric potential picture. What is the minimum
kinetic energy of the trigger electron?
Question C
It is easy to produce a potential difference of several thousand volts between your body
and the floor by scuffing your shoes across a nylon carpet. When you touch a metal
doorknob, you get a mild shock. Yet contact with a power line of comparable voltage
would probably be fatal. Why is there a difference?
Question D
Advice to mountaineers caught in lightning and thunderstorms is to (i) get rapidly off of
peaks and ridges and (ii) put both feet together and crouch in the open, only the feet
touching the ground. What is the physics basis for this good advice?
Question E
(i) Accurately draw the field (both scalar equipotential and vector electric) lines between
a positive point charge and an infinite grounded conducting plane. Briefly explain how
your field lines correctly illustrate the nature of the field around the conducting plane. (ii)
By looking at your plot, write down the mathematical equation that describes the E-field
for this point charge in this situation. Hint: derive it by only looking at the shape of the
field.
Problem 1
It is often a useful approximation to treat a nucleus as a uniformly charged
sphere of radius R and charge Q.
a. Derive the electrostatic potential V(r) for (i) r > R (outside the nucleus) and (ii)
R > r (inside the nucleus). Hint: this was derived in lecture.
b. Estimate the electrostatic potential energy of a proton at the center of a (i) 12C
nucleus and (ii) 208Pb nuclei? Use the radius function R = R0A1/3 where R0 =
1.07 fm and A = atomic mass number of the nuclei. Useful hint: ke2 = 1.44
MeV∙fm.
c. The binding energy of electrons is typically of the order 10 eVs. Compare the
nuclear binding energy to that of electrons, how do they compare? Explain
why protons do not participate in chemical reactions in general?
Answer: kQ/r, (kQ/2R)(3−r2/R2); 5.29 MeV, 28 MeV
Problem 2
The nuclear potential (i.e. the binding energy of the nucleons) is reduced by the
electrostatic repulsion of the protons inside the nucleus.
If we continue to treat the nucleus as a uniformly charged sphere of charge Q
and radius R as in problem (1) above, we now want to calculate the total
electrostatic energy inside the nucleus. The electric potential V(r) for r < R was
derived in Problem 1.
a. Derive the charge element dq contained between radius r and r + dr, using
the relation
  constant 
dq Qtotal

dV Vtotal
where dV  4r 2dr
b. Show that the total Coulomb energy contained with a uniformly charged
nucleus is
UCoulomb 
1
2

R
0
V(r) dq 
3kQ2
5R
Problem 24.9
An infinite nonconducting sheet has a surface charge density σ = +5.80 pC/m2. (a) How
much work is done by the electric field due to the sheet if a particle of charge q=
+1.60×1019 C is moved from the sheet to a point P at distance d = 3.56 cm from the
sheet? (b) If the electric potential V is defined to be zero on the sheet, what is V at P?
Problem 24.19
In Fig. 24-40, particles with the charges q1=+5e and q2=−15e are fixed in place with a
separation of d = 24.0 cm. With electric potential defined to be V
0 at infinity, what are the finite (a) positive and (b) negative
values of x at which the net electric potential on the x axis is
zero?
Problem 24.57
Identical 50 μC charges are fixed on an x axis at x = ±3.0 m.A particle of charge q =
−15μC is then released from rest at a point on the positive part of the y axis. Due to the
symmetry of the situation, the particle moves along the y axis and has kinetic energy 1.2
J as it passes through the point x = 0, y = 4.0 m. (a) What is the kinetic energy of the
particle as it passes through the origin? (b) At what negative value of y will the particle
momentarily stop?
Problem 24.59
In Fig. 24-60, a charged particle (either an electron or a proton) is moving rightward
between two parallel charged plates separated by distance d = 2.00
mm. The plate potentials are V1=−70.0 V and V2=−50.0 V. The particle
is slowing from an initial speed of 90.0 km/s at the left plate. (a) Is the
particle an electron or a proton? (b) What is its speed just as it reaches
plate 2?
Problem 24.66
Two isolated, concentric, conducting spherical shells have radii R1 = 0.500 m and R2 =
1.00 m, uniform charges q1=+2.00 μC and q2=+1.00 μC, and negligible thicknesses.
What is the magnitude of the electric field E at radial distance (a) r = 4.00 m, (b) r =
0.700 m, and (c) r = 0.200 m? With V = 0 at infinity, what is V at (d) r = 4.00 m, (e) r =
1.00 m, (f) r = 0.700 m, (g) r = 0.500 m, (h) r = 0.200 m, and (i) r = 0? ( j) Sketch E(r)
and V(r).
Problem 24.67
A metal sphere of radius 15 cm has a net charge of 3.0 ×108 C. (a) What is the electric
field at the sphere’s surface? (b) If V = 0 at infinity, what is the electric potential at the
sphere’s surface? (c) At what distance from the sphere’s surface has the electric
potential decreased by 500 V?
Problem 24.95
A thick spherical shell of charge Q and uniform volume charge density r is bounded by
radii r1 and r2 > r1. With V = 0 at infinity, find the electric potential V as a function of
distance r from the center of the distribution, considering regions (a) r > r2, (b) r2 > r >
r1, and (c) r < r1. (d) Do these solutions agree with each other at r = r2 and r = r1? (Hint:
See Module 23-6.)