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# HW7 - cylindar 4 An annular region b< ρ< a is bounded...

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Electromagnetics HW.7 Group.3 Deadline 30 / 8 / 89 1. Consider a rectangular box which is bounded by the planes y = 0, y = d , x = 0, x = a and extending to infinty in z direction, assume that the boundary conditions are: x = 0 : V = V 0 x = a : ∂V ∂n = 0 y = 0, y = d : ∂V ∂n = 0 find the electrical potential inside the region. 2. The potential distribution is to be determined in a region bounded by the planes y = 0 and y = d and extending to infinity in the x and z directions. In this region, there is a uniform charge density ρ 0 . On the upper boundary, the potential is Φ( x, d, z ) = V a sin( βx ). On the lower boundary, the potential is Φ( x, 0 , z ) = V b sin( αx ). Find the potential function throughtout the region 0 < y < d . 3. Suppose that a cylindar with radius R has a surface charge density of σ = σ 1 sin 2 φ + σ 2 cos φ . Find the potential everywhere(inside and outside the
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Unformatted text preview: cylindar). 4. An annular region b < ρ < a is bounded from outside at ρ = a by a surface having the potetial Φ = V a cos 3 φ and from the inside at ρ = b by a surface having a potential Φ = V b sin φ . Show that Φ in the annulus can be written as the sum of two terms, each a combination of solutions to Laplace’s equation designed to have the correct value at one radius while being zero at the other.(i.e, supperposition method). 5. A long dielectric cylindar of radius b and dielectric constant ε r is placed along the z axis in an initially uniform electric ﬁeld ⃗ E = E ˆ x . Determine Φ( ρ,φ ) and ⃗ E ( ρ,φ ) both inside and outside the dielectric cylindar. 1...
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