Chapter 18

The paper dielectric in a paper-and-foil capacitor is 0.0800 mm thick. Its dielectric constant is \(2.50,\) and its dielectric strength is 50.0 \(\mathrm{MV} / \mathrm{m}\) . Assume that the geometry is that of a parallel- plate capacitor, with the metal foil serving as the plates. (a) What area of each plate is required for a 0.200\(\mu F\) capacitor? (b) If the electric field in the paper is not to exceed one-half the dielectric strength, what is the maximum potential difference that can be applied across the capacitor?

(a) If a spherical raindrop of radius 0.650 \(\mathrm{mm}\) carries a charge of \(-1.20 \mathrm{pC}\) uniformly distributed over its volume, what is the potential at its surface? (Take the potential to be zero at an infinite distance from the raindrop. (b) Two identical raindrops, each with radius and charge specified in part (a) collide and merge into one larger raindrop. What is the radius of this larger drop, and what is the potential at its surface, if its charge is uniformly distributed over its volume?

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are 4.98 V and \(12.0 \mathrm{V} / \mathrm{m},\) respectively. (Take the potential to be zero at infinity.) (a) What is the distance to the point charge? (b) What is the magnitude of the charge? (c) Is the electric field directed toward or away from the point charge?

Two oppositely charged identical insulating spheres, each 50.0 \(\mathrm{cm}\) in diameter and carrying a uniform charge of magnitude \(175 \mu \mathrm{C},\) are placed 1.00 \(\mathrm{m}\) apart center to center Fig. 18.53 ). (a) If a voltmeter is connected between the nearest points \((a\) and \(b)\) on their surfaces, what will it read? (b) Which point, \(a\) or \(b,\) is at the higher potential? How can you know this without any calculations?

Potential in human cells. Some cell walls in the human body have a layer of negative charge on the inside surface and a layer of positive charge of equal magnitude on the outside surface. Suppose that the charge density on either surface is \(\pm 0.50 \times 10^{-3} \mathrm{C} / \mathrm{m}^{2},\) the cell wall is 5.0 \(\mathrm{nm}\) thick, and the cell-wall material is air. (a) Find the magnitude of \(\vec{E}\) in the wall between the two layers of charge. (b) Find the potential difference between the inside and the outside of the cell. Which is at the higher potential? (c) A typical cell in the human body has a volume of \(10^{-16} \mathrm{m}^{3} .\) Estimate the total electric-field energy stored in the wall of a cell of this size. (Hint: Assume that the cell is spherical, and calculate the volume of the cell wall.) (d) In reality, the cell wall is made up, not of air, but of tissue with a dielectric constant of \(5.4 .\) Repeat parts (a) and (b) in this case.

An alpha particle with a kinetic energy of 10.0 MeV makes a head-on collision with a gold nucleus at rest. What is the distance of closest approach of the two particles? (Assume that the gold nucleus remains stationary and that it may be treated as a point charge. The atomic number of gold is \(79,\) and an alpha particle is a helium nucleus consisting of two protons and two neutrons.)

In the Bohr model of the hydrogen atom, a single electron revolves around a single proton in a circle of radius \(r .\) Assume that the proton remains at rest. (a) By equating the electric force to the electron mass times its acceleration, derive an expression for the electron's speed. (b) Obtain an expression for the electron's kinetic energy, and show that its magnitude is just half that of the electric potential energy. (c) Obtain an expression for the total energy, and evaluate it using \(r=\) \(5.29 \times 10^{-11} \mathrm{m} .\) Give your numerical result in joules and in electron volts.

A proton and an alpha particle are released from rest when they are 0.225 nm apart. The alpha particle (a helium nucleus) has essentially four times the mass and two times the charge of a proton. Find the maximum speed and maximum acceleration of each of these particles. When do these maxima occur, just following the release of the particles or after a very long time?

A parallel-plate air capacitor is made from two plates 0.200 m square, spaced 0.800 \(\mathrm{cm}\) apart. It is connected to a \(120-\mathrm{V}\) battery. (a) What is the capacitance? (b) What is the charge on each plate? (c) What is the electric field between the plates? (d) What is the energy stored in the capacitor? (e) If the battery is disconnected and then the plates are pulled apart to a separation of \(1.60 \mathrm{cm},\) what are the answers to parts (a), (b), \((c),\) and \((d) ?\)

A capacitor consists of two parallel plates, each with an area of 16.0 \(\mathrm{cm}^{2}\) , separated by a distance of 0.200 \(\mathrm{cm} .\) The material that fills the volume between the plates has a dielectric constant of \(5.00 .\) The plates of the capacitor are connected to a \(300-\mathrm{V}\) battery. (a) What is the capacitance of the capacitor? (b) What is the charge on either plate? (c) How much energy is stored in the charged capacitor?

Electronic flash units for cameras contain a capacitor for storing the energy used to produce the flash. In one such unit, the flash lasts for \(\frac{1}{675}\) s with an average light power output of \(2.70 \times 10^{5} \mathrm{W}\) (a) If the conversion of electrical energy to light is 95\(\%\) efficient (the rest of the energy goes to thermal energy), how much energy must be stored in the capacitor for one flash? (b) The capacitor has a potential difference between its plates of 125 \(\mathrm{V}\) when the stored energy equals the value calculated in part (a). What is the capacitance?

A parallel-plate capacitor is made from two plates 12.0 \(\mathrm{cm}\) on each side and 4.50 mm apart. Half of the space between these plates contains only air, but the other half is filled with Plexiglas\oplus of dielectric constant 3.40 . (See Figure \(18.55 .\) An 18.0 V battery is connected across the plates. (a) What is the capacitance of this combination? (Hint: Can you think of this capacitor as equivalent to two capacitors in parallel? (b) How much energy is stored in the capacitor? (c) If we remove the Plexiglas@, but change nothing else, how much energy will be stored in the capacitor?

A parallel-plate capacitor with plate separation \(d\) has the space between the plates filled with two slabs of dielectric, one with constant \(K_{1}\) and the other with constant \(K_{2, \text { and each }}\) having thickness \(d / 2\) . (a) Show that the capacitance is given by \(C=\frac{2 \epsilon_{o} A}{d}\left(\frac{K_{1} K_{2}}{K_{1}+K_{2}}\right) \cdot(\)Hint: Can you think of this combination as two capacitors in series? (b) To see if your answer is reasonable, check it in the following cases: (i) There is only one dielectric, with constant \(K,\) and it completely fills the space between the plates. (ii) The plates have nothing but air, which we can treat as vacuum, between them.

BIO The electric egg. The eggs of many species undergo a rapid change in the electrical potential difference across the outer membrane when they are fertilized. This change in potential difference affects the physiological development of the eggs. The poterntial difference across the membrane is called the membrane potential, \(V_{m},\) defined as the inside potential minus the outside potential. The membrane potential \(V_{m}\) arises when protein enzymes use the energy available in ATP to actively expel sodium ions (Na') and accumulate potassium ions \(\left(\mathrm{K}^{+}\right) .\) Because the membrane of the unfertilized egg is selectively permeable to \(\mathrm{K}^{+},\) the \(V_{m}\) of the resting sea urchin egg is about \(-70 \mathrm{mV}\) ; that is, the inside has a potential of 70 \(\mathrm{mV}\) less than that of the outside. The egg membrane behaves as a capacitor with a specific capacitance of about 1\(\mu \mathrm{F} / \mathrm{cm}^{2} .\) When a sea urchin egg is fertilized, Na' channels in the membrane are opened, \(\mathrm{Na}^{+}\) enters the egg, and \(V_{m}\) rapidly changes to \(+30 \mathrm{mV},\) where it remains for several minutes. The concentration of \(\mathrm{Na}^{+}\) in the egg's interior is about 30 mmoles/liter (30 \(\mathrm{mM} )\) and 450 \(\mathrm{mM}\) in the surrounding sea water. The inside \(\mathrm{K}^{*}\) concentration is about 200 \(\mathrm{mM}\) and the outside \(\mathrm{K}^{+}\) is 10 \(\mathrm{mM} .\) A useful constant that connects electrical and chemical units is the Faraday number, which has a value of approximately \(10^{5}\) coulomb/mole. That is, an Avogadro number (a mole) of monovalent ions such as Na^ + or \(\mathrm{K}^{+}\) carries a charge of \(10^{5} \mathrm{C}\) . How many moles of \(\mathrm{Na}^{+}\) must move per unit area of membrane to change \(V_{m}\) from \(-70 \mathrm{mV}\) to \(+30 \mathrm{mV},\) making the assumption that the membrane behaves purely as a capacitor? A. \(10^{-4}\) mole \(/ \mathrm{cm}^{2}\) B. \(10^{-9}\) mole/cm \(^{2}\) C. \(10^{-12} \mathrm{mole} / \mathrm{cm}^{2}\) D. \(10^{-14} \mathrm{mole} / \mathrm{cm}^{2}\)