Electric Fields

8-1

(a) Two protons in a molecule are separated by a distance of $3.80\times 10^{-10} \rm m$. Find the electric force exerted by one proton on the other. (b) How does the magnitude of this force compare with the magnitude of the gravitational force between two protons? (c) What must be the charge-to-mass ratio of a particle if the magnitude of the gravitation force between two of these particles equals the magnitude of the electric force between them?

8-2

In the Bohr theory of the hydrogen atom, an electron moves in a circular orbit around a proton, where the radius of the orbit is $0.529\times 10^{-10} \rm m$. (a) Find the electric force between the two. (b) If this force causes the centripetal acceleration of the electron, what is the speed of the electron?

8-3

Figure 12a shows two charges, $q_1$ and $q_2$, held at a fixed distance $d$ apart. (a) What is the strength of the electric force that acts on $q_1$? Assume that $q_1 = q_2 = 20 \rm\mu C$ and $d = 1.5 \rm m$ (b) A third charge $q_3 = 20 \rm\mu C$ is brought in and placed as shown in Fig. 12b. What now is the strength of the electric force on $q_1$?

Figure 12: Prob 8-3

8-4

Three charged particles lie on a straight line and are separated by a distance $d$ as shown in Fig. 13. Charges $q_1$ and $q_2$ are held fixed. Charge $q_3$, which is free to move, is found to be in equilibrium under the action of the electric forces. Find $q_1$ in terms of $q_2$.

Figure 13: Prob 8-4

8-5

An airplane is flying through a thundercloud at a height of 2000 m (This is a very dangerous thing to do because of updrafts, turbulence and the possibility of electric charge). If there are charge concentrations of $+40.0 \rm C$ at a height of 3000 m within the cloud and of $-40.0 \rm C$ at a height of 1000 m, what is the electric field $E$ at the aircraft?

8-6

$2.00 \rm\mu C$ point charges are located on the $x$ axis. One is at $x=1.00 \rm m$, and the other is at $x=-1.00 \rm m$. (a) Determine the electric field on the $y$ axis at $y=0.500 \rm m$. (b) Calculate the electric force on an $-3.00 \rm\mu C$ charge placed on the $y$ axis at $y=0.500 \rm m$.

8-7

Calculate the electric field, magnitude and direction, due to an electric dipole, at a point $P$, located at a distance $r\gg d$ along the perpendicular bisector of the line joining the charges; see Fig. 14. Express your answer in terms of the electric dipole moment $\bf p$.

Figure 14: Prob 8-7

8-8

A continuous line of charge lies along the $x$ axis, extending from $x=+x_0$ to positive infinity. The line carries a uniform linear charge density $\lambda _0$. What are the magnitude and direction of the electric field at the origin?

8-9

The electric field along the axis of a uniformly charged disc of a radius $R$ and total charge $Q$ was calculated in the textbook. Show that the electric field at distances $x\gg R$ approaches that of a point charge $Q=\sigma \pi R^2$ where $\sigma$ is the charge per unit area. (Hint: first show that ${x/ {(x^2 + R^2)}^{1/2}} = (1+R^2/x^2)^{-1/2}$, and use the binomial expansion $(1+\delta )^n \approx 1+n\delta$ when $\delta \ll 1$.)

8-10

A negatively charged rod of finite length has a uniform charge per unit length. Sketch the electric field lines in a plane containing the rod.

8-11

A proton is projected in the positive $x$ direction into a region of uniform electric field ${\bf E}=-6.00\times 10^{5}{\bf i} \rm N/C$. The proton travels 7.00 cm before coming to rest. Determine (a) the acceleration of the proton, (b) its initial speed, and (c) the time it takes the proton to come to rest.

8-12

The electrons in a particle beam each have a kinetic energy of $1.60\times 10^{-17}  \rm J$. What are the magnitude and direction of the electric field that stops these electrons in a distance of 10.0 cm?