87. Review. An electric dipole in a uniform horizontal electric field is displaced slightly from ts equilibrium position as shown in Figure P23.87, where θ is small The separation of the charges is 2a, and each of the two particles has mass m. (a) Assuming the dipole is released from this position, show that its angular ori entation exhibits simple harmonic motion with a frequency What If (b) Suppose the masses of the two charged particles in the dipole are not the same even though each particle continues to have charge q. Let the masses of the particles be m and m2. Show that the fre quency of the oscillation in this case is E mm 2amm2 Figure P23.87

A proton and an alpha particle (charge = 2e, mass = 6.64 1027 kg) are initially at rest, separated by 4.00 1015 m. (a) If they are both released simultaneously, explain why you cant find their velocities at infinity using only conservation of energy. (b) What other conservation law can be applied in this case? (c) Find the speeds of the proton and alpha particle, respectively, at infinity.

A Van de Graaff generator is charged so that a proton at its surface accelerates radially outward at 1.52 1012 m/s3. Find (a) the magnitude of the electric force on the proton at that instant and (b) the magnitude and direction of the electric field at the surface of the generator.

An electric dipole is located along the y axis as shown in Figure P24.48. The magnitude of its electric dipole moment is defined as p = 2aq. (a) At a point P, which is far from the dipole (r a), show that the electric potential is V=kepcosr2 (b) Calculate the radial component Er and the perpendicular component E of the associated electric field. Note that E = (1/r)(V/). Do these results seem reasonable for (c) = 90 and 0? (d) For r = 0? (e) For the dipole arrangement shown in Figure P24.48, express V in terms of Cartesian coordinates using r = (x2 + y2)1/2 and cos=y(x2+y2)1/2 (f) Using these results and again taking r a, calculate the field components Ex and Ey. Figure P24.48