HW5 Solutions Notice numbers may change randomly in your assignments and you may have to recalculate solutions for your specific case. Tipler 21.P.018 Estimate the force required to bind the two protons in the He nucleus together. (Hint: Model the protons as point charges. Assume the diameter of the He nucleus to be approximately 10-15 m.) Solution: Since the nucleus is in equilibrium:

!

Fbinding = Felectrostatics " Fbinding =kq

2

r2

=9 #10

9# 1.6 #10

$19( )2

10$15( )

2= 0.23kN

Tipler 21.P.023 What is the total charge of all the protons in 1.50 kg of carbon? Solution: Carbon has Z = 6 protons hence Q = 6nCe, where nc = the number of atoms in m = 0.20 kg of carbon. Since NA = Avogadro number = number of Carbon atoms in a mole = number of atoms in M = 12 g of Carbon-12 = 6.022 x 1023 atoms/mol:

NA : M = nC : m

!

" nC

=mN

A

M" Q = 6e

mNA

M= 7.23#10

7C

Tipler 21.P.028 A 2.0 µC point charge and a 4.0 µC point charge are a distance L apart. Where should a third point charge be placed so that the electric force on that third charge is zero? Solution:

q3 q2 q4

x L-x

Tipler 21.P.035 Five identical point charges, each having charge Q, are equally spaced on a semicircle of radius R as shown in the figure below. Find the force (in terms of k, Q, and R) on a charge q located equidistant from the five other charges. (Use the following variables as necessary: Q, R, q, k, i for , and j for .) Solution:

Tipler 21.P.069

A rigid 1.00 m long rod is pivoted about its center. A charge q1 =

5.00 10-7 C is placed on one end of the rod, and a charge q2 = -

q1 is placed a distance d = 10.0 cm directly below it.

(a) What is the force exerted by q2 on q1?

(b) What is the torque (measured about the rotation axis) due to

that force?

(c) To counterbalance the attraction between the two charges,

we hang a block 25.0 cm from the pivot as shown. What value

should we choose for the mass m of the block? (d) We now move the block and hang it a distance of 25.0 cm from the balance point, on the same side of the balance as the charge. Keeping q1 the same, and d the same, what value should we choose for q2 to keep this apparatus in balance?

Solution:

Tipler 21.P.060

A dipole of moment 0.50 e · nm is placed in a uniform electric

field with a magnitude of 4.0 104 N/C. What is the magnitude of

the torque on the dipole for each of the following situations?

(a) the dipole is aligned with the electric field

(b) the dipole is transverse to (perpendicular to) the electric field

(c) the dipole makes an angle of 30.0° with the direction of the

electric field

(d) Defining the potential energy to be zero when the dipole is

transverse to the electric field, find the potential energy of the

dipole in the electric field for the orientations specified in Parts

(a) and (c). Solution:

Tipler 21.P.051

The acceleration of a particle in an electric field depends on q/m

(the charge-to-mass ratio of the particle).

(a) Compute q/m for an electron. (b) What are the magnitude and direction of the acceleration of an electron in a uniform electric field that has a magnitude of 100 N/C?

(c) Compute the time it takes for an electron placed at rest in a

uniform electric field that has a magnitude of 100 N/C to reach a

speed 0.01c. (When the speed of an electron approaches the

speed of light c, relativistic kinematics must be used to calculate

its motion, but at speeds of 0.01c or less, nonrelativistic

kinematics is sufficiently accurate for most purposes.)

(d) How far does the electron travel in that time? Solution:

Tipler 22.P.013 A uniform line charge that has a linear charge density λ = 3.5 nC/m is on the x axis between x = 0 to x = 5.0 m.

(a) What is its total charge?

(b) Find the electric field on the x axis at x = 6.0 m.

(c) Find the electric field on the x axis at x = 9.0 m.

(d) Find the electric field on the x axis at x = 250 m.

(e) Estimate the electric field at x = 250 m, using the

approximation that the charge is a point charge on the x axis at x

= 2.5 m.

(f) Compare your result with the result calculated in part (d) by

finding the ratio of the approximation to the exact result. To do

this, you will need to assume that the values given in this

problem statement are valid to more than two significant figures. Solution:

Tipler 22.P.015

A charge of 2.75 µC is uniformly distributed on a ring of radius

8.5 cm. Find the electric field strength on the axis at the

following locations.

(a) 1.2 cm from the center of the ring

(b) 3.6 cm from the center of the ring

(c) 4.0 m from the center of the ring

(d) Find the field strength at 4.0 m using the approximation that

the ring is a point charge at the origin.

(e) Compare your results for parts (c) and (d) by finding the ratio

of the approximation to the exact result. Solution:

Tipler 22.P.029

An electric field is = 300 N/C i for x > 0, 0 if x = 0 and E = -300

N/C i for x < 0. A cylinder of length 20 cm and radius 4.0 cm has

its center at the origin and its axis along the x axis such that one

end is at x = +10 cm and the other is at x = -10 cm.

(a) What is the flux through each end?

(b) What is the flux through the curved surface of the cylinder?

(c) What is the net outward flux through the entire cylindrical

surface?

(d) What is the net charge inside the cylinder? Solution:

Tipler 22.P.032

What is the electric flux through one side of a cube that has a

single point charge of -3.00 µC placed at its center? Hint: You do

not need to integrate any equations to get the answer. Solution:

Problems for Practice Tipler 21.P.012

Four charges are fixed in place at the corners of a square as

shown below. No other charges are nearby. Which of the

following statements is true?

(a) E is zero midway between the top two charges and midway

between the bottom two charges

(b) E is zero at the center of the square (c) E is zero at the midpoints of all four sides of the square

Solution: (b) At the center of the square the 2 positive charges alone produce a net E-field of zero, as well as the 2 negative charges. Thus, the net force acting on a test charge at the midpoint of the square is zero. Tipler 21.P.027 Three point charges are on the x axis: q1 = -6.0 µC is at x = -3.0 m, q2 = 4.0 µC is at the origin, and q3 = -6.0 µC is at x = 3.0 m. Find the force on q1. Solution:

Tipler 22.P.016

A nonconducting disk of radius R lies in the z = 0 plane with its

center at the origin. The disk has a uniform surface charge

density σ. Find the value of z for which Ez = σ/(4ε0). Note that at

this distance, the magnitude of the electric field strength is half

the electric field strength at points on the z axis that are very

close to the disk. (Use R as necessary.)

Solution: