2. Derive an expression for ... charges together as indicated in Fig. 28-28 below. Each side... Homework #4 203-1-1721 ...

2. Derive an expression for ... charges together as indicated in Fig. 28-28 below. Each side... Homework #4 203-1-1721 ...

Homework #4
203-1-1721
Physics 2 for Students of Mechanical Engineering
Part A
2. Derive an expression for the work required by an external agent to put the four
charges together as indicated in Fig. 28-28 below. Each side of the square as length
a.
6. Two parallel, flat, conducting surfaces of spacing d = 1.0 cm have a potential
difference V of 10.3 kV. An electron is projected (launched) from one plate
directly toward the second. What is the initial velocity (vi) of the electron if it comes
to rest just at the surface of the second plate?
10. An electron is projected with an initial speed of vi = 3.44 x 105 m/s directly toward
a proton that is essentially at rest. If the electron is initially a great distance from the
proton, at what distance from the proton is its speed instantaneously equal to twice
its initial value (i.e., vf = 2vi).
14. An infinite sheet of charge has a charge density = 0.12 x 10-6 C/m2. How far apart
are the equipotential surfaces whose potentials differ by 48 V?
18. Compute the escape speed for an electron from the surface of a uniformly charged
sphere of radius 1.22 cm and total charge +1.76 x 10-15 C. Neglect gravitational
forces.
25. (a) For Fig. 28-34 below, derive an expression for V = VA - VB. (b) Consider the
following limiting cases: Does your result reduce to the expected answer when d =
0? When a = 0? When q = 0?
30. Suppose that the electric potential varies along the x axis as shown in the graph of
Fig. 28-37 below. Of the intervals shown (ignore the behavior at the endpoints of the
intervals), determine the intervals in which Ex has (a) its greatest absolute value and
(b) its least absolute value. (c) Plot Ex versus x.
37. In moving from A to B along an electric field line, the electric field does 3.94 x
10 19 J of work on an electron in the field illustrated in Fig. 28-38 below. What are
the following differences in the electric potential (a) VB - VA, (b) VC - VA, and (c) VC VB?
Part B
4. The electric field inside a nonconducting sphere of radius R, containing a uniform
charge density, is radially directed and has magnitude E = (qr)/(4 oR3), where q is
the total charge in the sphere and r is the distance form the center of the sphere. (a)
Find the potential V inside the sphere, taking V = 0 at r = 0. (b) What is the
difference in electric potential V between a point on the surface and the center of
the sphere? If q > 0, which point is at the higher potential? (c) Show that the
potential at a distance r from the center, where r < R, is given by V = q(3R2 –
r2)/(8 oR3), where the zero of the potential is taken at r = (infinity). Why does
this result differ from that of part (a)?
7. A spherical drop of water carrying a charge of +32 x 10-12 C has a potential of 512 V
at its surface. (a) What is the radius of the drop? (b) If two such drops of the same
charge and radius combine to form a single spherical drop, what is the potential at
the surface of the new drop? Set V = 0 at infinity.
8. Figure 28-42 below shows, edge-on, an "infinite" sheet of positive charge density .
(a) How much work is done by the electric field of the sheet as a small positive test
charge qo is moved from an initial position on the sheet to a final position located at
a perpendicular distance z from the sheet? (b) Use the result from (a) to show that
the electric potential of an infinite sheet of charge can be written V = Vo – ( /2 o)z,
where Vo is the potential at the surface of the sheet.
12. A charge per unit length is distributed uniformly along a thin rod of length L. (a)
Determine the potential (chosen to be zero at infinity) at point P a distance y from
one end of the rod and in line with it (see Fig. 28-45 below). (b) Use the result of (a)
to compute the component of the electric field at P in the y direction (along the rod).
(c) Determine the component of the electric field at P in a direction perpendicular to
the rod.
14. Two identical conducting spheres of radius 15.0 cm are separated by a distance of
10.0 m. What is the charge on each sphere if the potential of one is +1500 V and the
other is -1500 V? What assumptions have you made? Take V = 0 at infinity.
Homework #4 Solutions 203-1-1721 Physics 2 for Students of Mechanical Engineering
Part A
Part B
Erratum (Corrections to homework)
Homework #4: P28-8
G G
qσz
(a) The work done by the field is W = ∫ F ⋅ dr = + Fz = + Eqz = +
.
2ε 0
(b) Since the work done by the field is
σz
σz
W = −q∆V , then ∆V = −
and V = V0 −
.
2ε 0
2ε 0
∂V
σ
=+
,
∂z
2ε 0
which confirms that the electric field points in the positive z-direction, as it must.
Note that
E=−