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Unformatted text preview: Electromagnetic Theory I Solution Set 2 Due: 7 September 2011 5. a) Prove that an electron placed at the center of a spherical cavity in a conductor is shielded from external gravitational fields. In other words the total force on that electron is zero when the conductor is held fixed in an arbitrary (static and Newtonian) gravitational field. Solution : Since the mobile charge carriers are electrons, static equilibrium is achieved when the total force on each electron, in the body of the conductor, F = − e E + m e g = −∇ ( − eφ e + m e φ g ) = 0. Thus ( − eφ e + m e φ g ) = C , a constant throughout the body of the conductor. By uniqueness of solutions of Laplace’s equation, ( − eφ e + m e φ g ) = C also in empty hollow cavities within the conductor. Since we place an electron at the center of a spherical cavity, the further charge rearrangement in the conductor due to its field will be uniformly distributed on the spherical wall and will exert no net force. Thus the electron will feel the force − e E + m e g = 0. b) Prove that the instantaneous acceleration of a similarly placed positron (with charge equal and opposite to that of the electron and masss equal to that of the electron) is twice the normal gravitational acceleration. Solution: A positron has charge + e so the force on it due to the external field will be + e E + m e g = 2 m e g so its instantaneous acceleration is 2 g . c) Find the instantaneous acceleration of a similarly placed muon, which has the same charge as the electron but a different mass m μ > m e . (In fact the muon is about 200 times heavier than the electron). Solution: A muon has charge − e so the force on it due to the external field will be − e E + m μ g = ( m μ − m e ) g so its instantaneous acceleration is (1 − m e /m μ ) g . 6. Four identical spherical conductors are placed with their centers at the corners of a square. Number them 1,2,3,4 as you go clockwise around the square. Consider the capacitor coefficients C ij for this system....
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