Express the net capacitance

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Reference no: EM133206530

Case: We seek the capacitance per unit length of a circular conducting wire placed over the middle of a conducting strip. The wire radius is a, and the center of the wire is located at a height H over the strip. The strip has width w and thickness b. The geometry is assumed to be two-dimensional and the structure is located in free space. For the computation of the capacitance per unit, it is therefore appropiate to use the method of moments.

q₁ = C₁₁V₁ + C₁₂V₂

For a capacitor where two plates have different shape, the two plates must have opposite total charges. However, they will not in general have opposite potentials. Since Poisson's equation is linear, the charges and potentials must satisfy the linear relation

where q_i are the charges on the plates and V_i their potentials. The matrix elements C_ij of the capacitance matrix are referred to as capacitance coefficients. Note that the capacitance coefficients in themselves are not physically very relevant and they depend on how the normalizing distance is chosen in the expression for the potential from a line charge. The coefficients can be computed by considering two cases where V₁ and V₂ take linearly independent values, e.g.,

1. V₁ = 1 and V₂ = 0 yield the values for C₁₁ and C₂₁.

2. V₁ = 0 and V₂ = 1 yield the values for C₁₁ and C₂₂.

To define the capacitance, we require that the net charge is zero. This implies that q₁ = -q₂ = q. Then the capacitance is defined as

C = q/V₁-V₂

Question 1: First express the net capacitance C analytically in terms of the capacitance coefficients C₁₁, C₁₂, C₂₁ and C₂₂.

Question 2: Place your answer in this markdown cell.

Reference no: EM133206530

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