110Amid1+solns

110Amid1+solns - R θ = V(3 cos 2 θ − 1 where V is a...

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Midterm 1 Physics 110A 1/26/11 Please show your work and explain your answers. Credit will not be given for writing down an answer with an invalid or missing explanation even if your Fnal answer is correct. 1. (20 points) A long metal cylinder of radius a is surrounded by a concentric cylindrical metal shell of inner radius b . See the diagram below. A potential diFerence is applied between the inner and outer cylinder. Calculate the resulting capacitance per unit length. a b 2. (20 points total) In the following r is the position vector : r = x ˆx + y ˆy + z ˆ z . Calculate: (a) (7 points) What is ∇ · r ? (b) (7 points) What is ∇ × r ? (c) (6 points)What is 2 ( r · ˆx )? 3. (20 points) Consider a semicircular bowl, or hemisphere of radius r , facing upside down, so that it is resting on the x y plane. What is the total ±ux passing through the surface of this bowl, when a uniform electric ²eld E is applied in the z direction? 4. (30 points) The potential on the surface of a hollow sphere of radius R is V (
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Unformatted text preview: R, θ ) = V (3 cos 2 ( θ ) − 1), where V is a constant. Calculate the potential inside the sphere as a function of position r and angle θ . 5. (30 points) Two semi in²nite grounded conducting regions meet at right angles. In the region between them, there is a point charge q , at a position ( a, b ), situated as shown below. The grey region is conductor and at potential V = 0. The conductor extends in²nitely both into and out of the page. Set up the image con²guration and calculate the potential in this region without conductor. y x V=0 a b q Additional Information: P ( x ) = 0 , P 1 ( x ) = x, P 2 ( x ) = 3 x 2 − 1 2 , P 3 ( x ) = 5 x 3 − 3 x 2 Basic equations for electrostatics: F = q E , ∇ · E = ρ ǫ , E = −∇ V, ∇ × E = 0 In spherical coordinates ∇ 2 V = 1 r 2 ∂ ∂r ( r 2 ∂V ∂r ) + 1 r 2 sin θ ∂ ∂θ (sin θ ∂V ∂θ ) + 1 r 2 sin 2 θ ∂ 2 V ∂φ 2 ....
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This note was uploaded on 09/25/2011 for the course PHYS 110a taught by Professor Johnson,r during the Winter '08 term at UCSC.

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110Amid1+solns - R θ = V(3 cos 2 θ − 1 where V is a...

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