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Solution to HW3

# Solution to HW3 - Problem 1 a Boundary conditions include...

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Problem 1 a) Boundary conditions include continuity of potential and electrical field (derivative of potential) at x = 0, and continuity of electrical field at x = - x p and x = x n , namely 1. V p (0) = V n (0) 2. ∂x V p (0) = ∂x V n (0) 3. ∂x V p ( - x p ) = 0 4. ∂x V n ( x n ) = 0 We also have the constraint that V n ( x n ) - V p ( - x p ) = V bi . This gives rise to the dependence of parameters x n , x p and V bi which is asked to be found in part d). b) The charge density within each region is constant. By solving the Poisson’s equation, we have that the potential within each region is a quadratic function with respect to x . V p ( x ) = eN A 2 K s ε 0 x 2 + Ax + B V n ( x ) = - eN D 2 K s ε 0 x 2 + Cx + D By boundary condition 1, we have

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