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Learn how to solve electrostatic problems. Overview of solution methods. Simple 1-D problems. Reduce Poisson’s equation to Laplace’s equation. Capacitance. The method of images. Overview. Illustrated below is a fairly general problem in electrostatics. Many practical problems are special cases of this general problem. Where to start?
Examples of Electrostatic Problems with Dielectrics Problem: Find (electric flux density), (electric field intensity), and (polarization) for a metallic sphere (radius a, charge Q), coated by a dielec-tric (radius b), and the charge densities at the interfaces. Solution: Use Gauss’ Law In region 0, In region 1, a < r < b:
So for V, we have only one 2 nd order DE to solve, but if we approach the problem using electric field q, we end up with two equations: · q L Ù, H q L Ù In general, there are two major ways to solve the potential problems in electrostatic: (a) Solve as a source problem using integration, (b) Solve as a boundary value problem, using boundary
The problems statement implies V0 is complex-valued. To accomodate this, we define the magnitude and phase of V0 as follows: V0,|V0|ejπ/3 Then: v(x,t) = Re |V0|ejπ/3e+jβxejωt = |V0|Re ej(ωt +βx π/3) Finally, using the identity ejθ= cosθ+jsinθ, we obtain v(x,t) = |V0|cos(ωt+βx+π/3) This wave is traveling in the −xdirection. 12
Imaginary dipoles. Consider two point charges q and q0 at a distance d from each other, and a plane perpen-dicular to the line through q and q0 in a distance d from q. Show that, in order for the plane to be at constant potential, one must have q0 = q and = 1=2. Hint: Look at the potential at large distance.
To calculate the force exerted by some electric charges, q1, q2, q3, ... (the source charges) on another charge Q (the test charge) we can use the principle of superposition. This principle states that the interaction between any two charges is completely unaffected by the presence of other charges. The force exerted on Q by q1, q2, and q3 (see ...
Coulomb’s Law: Problems and Solutions. Compute the electric force between two charges of 5×10−9 C and −3×10−8 C which are separated by d = 10 cm. Solution: the magnitude of the electrostatic force between two point charges is given by Coulomb’s law as. =.
