Homework problems
1 
In our bynowfamiliar neuron, the voltage difference between the inner and outer surfaces of the cell membrane is about V_{in}  V_{out} = 70 mV in the resting state, and the thickness of the membrane is about 6.0 nm
(i.e. only about a hundred atoms thick). What is the electric field inside the membrane?
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2  (a) The gap between the electrodes in an automobile engine's spark plug is 0.060 cm. To produce an electric spark in a gasolineair mixture, an electric field of 3.0x10^{6} V/m must be achieved. On starting a car, what minimum voltage must be supplied by the ignition circuit? Assume the field is constant. (b) The small size of the gap between the electrodes is inconvenient because it can get blocked easily, and special tools are needed to measure it. Why don't they design spark plugs with a wider gap?
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3  (a) At time t=0, a small, positively charged object is placed, at rest, in a uniform electric field of magnitude E. Write an equation giving its speed, v, in terms of t, E, and its mass and charge m and q.
(b) If this is done with two different objects and they are observed to have the same motion, what can you conclude about their masses and charges? (For instance, when radioactivity was discovered, it was found that one form of it had the same motion as an electron in this type of experiment.)
 
4  Show that the magnitude of the electric field produced by a simple twocharge dipole, at a distant point along the dipole's axis, is to a good approximation proportional to D/r^{3}, where r is the distance from the
dipole. [Hint: Use the approximation (1+ε)^{p}
≈ 1+pε , which is valid for small ε.]
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5  Given that the field of a dipole is proportional to D/r^{3} (see previous problem), show that its voltage varies as D/r^{2}. (Ignore positive and negative signs and numerical constants of proportionality.)
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6  A carbon dioxide molecule is structured like OCO, with all three atoms along a line. The oxygen atoms grab a little bit of extra negative charge, leaving the carbon positive. The molecule's symmetry, however, means that it has no overall dipole moment, unlike a Vshaped water molecule, for instance. Whereas the voltage of a dipole of magnitude D is proportional to D/r^{2} (see previous problem), it turns out that the voltage of a carbon dioxide molecule along its axis equals k/r^{3}, where r is the distance from the molecule and k is a constant. What would be the electric field of a carbon dioxide molecule at a distance r?
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7  A proton is in a region in which the electric field is given by E=a+bx^{3}. If the proton starts at rest at x_{1}=0, find its speed, v, when it reaches position x_{2}. Give your answer in terms of a, b, x_{2}, and e and m, the charge and mass of the proton.
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8  Consider the electric field created by a uniform ring of total charge q and radius b. (a) Show that the field at a point on the ring's axis at a distance a from the plane of the ring is kqa(a^{2}+b^{2}) ^{3/2}. (b) Show that this expression has the right behavior for a=0 and for a much greater than b.
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9  Consider the electric field created by an infinite uniformly charged plane. Starting from the result of problem 8, show that the field at any point is 2πkσ, where σ is the density of charge on the plane, in units of coulombs per square meter. Note that the result is independent of the distance from the plane. [Hint: Slice the plane into infinitesimal concentric rings, centered at the point in the plane closest to the point at which the field is being evaluated. Integrate the rings' contributions to the field at this point to find the total field.]
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10  Consider the electric field created by a uniformly charged cylinder which extends to infinity in one direction. (a) Starting from the result of problem 8, show that the field at the center of the cylinder's mouth is 2πkσ, where σ is the density of charge on the cylinder, in units of coulombs per square meter. [Hint: You can use a method similar to the one in problem 9.] (b) This expression is independent of the radius of the cylinder. Explain why this should be so. For example, what would happen if you doubled the cylinder's radius?
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11 
Three charges are arranged on a square as shown. All three charges are positive. What value of q_{2}/q_{1} will produce zero electric field at the center of the square?  S 
