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2- show that if √XÃ = 0 and VXB = 0, then V. (AXB) = 0

2- show that if √XÃ = 0 and VXB = 0, then V. (AXB) = 0

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### Problem Set **2.** Show that if \( \nabla \times \vec{A} = \vec{0} \) and \( \nabla \times \vec{B} = \vec{0} \), then \( \nabla \cdot (\vec{A} \times \vec{B}) = 0 \). **3.** A rectangle of sides \( a \) and \( b \) is placed in the x-y plane and centered at the origin with its sides parallel to the x- and y-axes. Assume the rectangle carries a uniform charge with charge density \( \rho_s \). - **a.** Use Coulomb's law to find \( \vec{E} \) at (0,0,z). - **b.** Find the electric potential \( V \) at (0,0,z). - **c.** Let \( a \to \infty \) and \( b \to \infty \), use part (a) to find the electric field due to an infinite sheet of charge.
Transcribed Image Text:### Problem Set **2.** Show that if \( \nabla \times \vec{A} = \vec{0} \) and \( \nabla \times \vec{B} = \vec{0} \), then \( \nabla \cdot (\vec{A} \times \vec{B}) = 0 \). **3.** A rectangle of sides \( a \) and \( b \) is placed in the x-y plane and centered at the origin with its sides parallel to the x- and y-axes. Assume the rectangle carries a uniform charge with charge density \( \rho_s \). - **a.** Use Coulomb's law to find \( \vec{E} \) at (0,0,z). - **b.** Find the electric potential \( V \) at (0,0,z). - **c.** Let \( a \to \infty \) and \( b \to \infty \), use part (a) to find the electric field due to an infinite sheet of charge.
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Step 1: Formula used

nabla times open parentheses A with rightwards arrow on top cross times B with rightwards arrow on top close parenthesesequals B with rightwards arrow on toptimes open parentheses nabla x A with rightwards arrow on top close parenthesesnegative A with rightwards arrow on top.left parenthesisnabla x B with rightwards arrow on top right parenthesis

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