### Additional Problems

A 12.0-V battery-operated bottle warmer heats 50.0 g of glass, $2.50\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}\phantom{\rule{0.2em}{0ex}}\text{g}$ of baby formula, and $2.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}\phantom{\rule{0.2em}{0ex}}\text{g}$ of aluminum from $20.0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$ to $90.0\phantom{\rule{0.2em}{0ex}}\text{\xb0C}$. (a) How much charge is moved by the battery? (b) How many electrons per second flow if it takes 5.00 min to warm the formula? (*Hint:* Assume that the specific heat of baby formula is about the same as the specific heat of water.)

A battery-operated car uses a 12.0-V system. Find the charge the batteries must be able to move in order to accelerate the 750 kg car from rest to 25.0 m/s, make it climb a $2.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}\text{-m}$ high hill, and finally cause it to travel at a constant 25.0 m/s while climbing with $5.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}\text{-N}$ force for an hour.

(a) Find the voltage near a 10.0 cm diameter metal sphere that has 8.00 C of excess positive charge on it. (b) What is unreasonable about this result? (c) Which assumptions are responsible?

A uniformly charged half-ring of radius 10 cm is placed on a nonconducting table. It is found that 3.0 cm above the center of the half-ring the potential is –3.0 V with respect to zero potential at infinity. How much charge is in the half-ring?

A glass ring of radius 5.0 cm is painted with a charged paint such that the charge density around the ring varies continuously given by the following function of the polar angle $\theta ,\lambda =\left(3.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-6}}\phantom{\rule{0.2em}{0ex}}\text{C/m}\right){\phantom{\rule{0.2em}{0ex}}\text{cos}}^{2}\theta .$ Find the potential at a point 15 cm above the center.

A CD disk of radius ($R=3.0\phantom{\rule{0.2em}{0ex}}\text{cm}$) is sprayed with a charged paint so that the charge varies continually with radial distance *r* from the center in the following manner: $\sigma =-\left(6.0\phantom{\rule{0.2em}{0ex}}\text{C/m}\right)r/R$.

Find the potential at a point 4 cm above the center.

(a) What is the final speed of an electron accelerated from rest through a voltage of 25.0 MV by a negatively charged Van de Graff terminal? (b) What is unreasonable about this result? (c) Which assumptions are responsible?

A large metal plate is charged uniformly to a density of $\sigma =2.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-9}}\phantom{\rule{0.2em}{0ex}}{\text{C/m}}^{2}$. How far apart are the equipotential surfaces that represent a potential difference of 25 V?

Your friend gets really excited by the idea of making a lightning rod or maybe just a sparking toy by connecting two spheres as shown in Figure 7.39, and making ${R}_{2}$ so small that the electric field is greater than the dielectric strength of air, just from the usual 150 V/m electric field near the surface of the Earth. If ${R}_{1}$ is 10 cm, how small does ${R}_{2}$ need to be, and does this seem practical? (*Hint:* recall the calculation for electric field at the surface of a conductor from Gauss’s Law.)

(a) Find $x\phantom{\rule{0.2em}{0ex}}>>\phantom{\rule{0.2em}{0ex}}L$ limit of the potential of a finite uniformly charged rod and show that it coincides with that of a point charge formula. (b) Why would you expect this result?

A small spherical pith ball of radius 0.50 cm is painted with a silver paint and then $\mathrm{-10}\phantom{\rule{0.2em}{0ex}}\mu \text{C}$ of charge is placed on it. The charged pith ball is put at the center of a gold spherical shell of inner radius 2.0 cm and outer radius 2.2 cm. (a) Find the electric potential of the gold shell with respect to zero potential at infinity. (b) How much charge should you put on the gold shell if you want to make its potential 100 V?

Two parallel conducting plates, each of cross-sectional area $400\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{2}$, are 2.0 cm apart and uncharged. If $1.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{12}$ electrons are transferred from one plate to the other, (a) what is the potential difference between the plates? (b) What is the potential difference between the positive plate and a point 1.25 cm from it that is between the plates?

A point charge of $q=5.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-8}}\phantom{\rule{0.2em}{0ex}}\text{C}$ is placed at the center of an uncharged spherical conducting shell of inner radius 6.0 cm and outer radius 9.0 cm. Find the electric potential at (a) $r=4.0\phantom{\rule{0.2em}{0ex}}\text{cm,}$ (b) $r=8.0\phantom{\rule{0.2em}{0ex}}\text{cm,}$ (c) $r=12.0\phantom{\rule{0.2em}{0ex}}\text{cm}\text{.}$

Earth has a net charge that produces an electric field of approximately 150 N/C downward at its surface. (a) What is the magnitude and sign of the excess charge, noting the electric field of a conducting sphere is equivalent to a point charge at its center? (b) What acceleration will the field produce on a free electron near Earth’s surface? (c) What mass object with a single extra electron will have its weight supported by this field?

Point charges of $25.0\phantom{\rule{0.2em}{0ex}}\mu \text{C}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}45.0\phantom{\rule{0.2em}{0ex}}\mu \text{C}$ are placed 0.500 m apart.

(a) At what point along the line between them is the electric field zero?

(b) What is the electric field halfway between them?

What can you say about two charges ${q}_{1}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{q}_{2}$, if the electric field one-fourth of the way from ${q}_{1}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}{q}_{2}$ is zero?

Calculate the angular velocity $\omega $ of an electron orbiting a proton in the hydrogen atom, given the radius of the orbit is $0.530\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-10}}\phantom{\rule{0.2em}{0ex}}\text{m}$. You may assume that the proton is stationary and the centripetal force is supplied by Coulomb attraction.

An electron has an initial velocity of $5.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{6}\phantom{\rule{0.2em}{0ex}}\text{m/s}$ in a uniform $2.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{-N/C}$ electric field. The field accelerates the electron in the direction opposite to its initial velocity. (a) What is the direction of the electric field? (b) How far does the electron travel before coming to rest? (c) How long does it take the electron to come to rest? (d) What is the electron’s velocity when it returns to its starting point?