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To find the density at (A(6, -2.5, 3)) given (V = 2x^2y + 20z – frac{4}{x^2 + y^2}), we use the fact that the electric field (E) is related to the electric potential (V) by (-vec{nabla} V), where (vec{nabla} V) represents the gradient of (V). However, since the question asks for density, it seems to be seeking the charge density. In electrostatics, the relationship between electric potential (V) and charge density (rho) is often given through Poisson’s equation:
[
nabla^2 V = -frac{rho}{epsilon_0}
]
where (nabla^2) is the Laplacian operator, and (epsilon_0) is the permittivity of free space ((8.85 times 10^{-12} , F/m) or (C^2/N cdot m^2)).
Given (V = 2x^2y + 20z – frac{4}{x^2 + y^2}), we need to first calculate the Laplacian (nabla^2 V) of this potential to find an expression for (rho).
The Laplacian (nab
Answer: a
Explanation: Find E from V, E = -Grad (V). We get E at A(6,-2.5,3) as 59.97i – 71.98j –
20k. Thus D = εE = 8.854 X 10-12 X
(59.97i – 71.98j -20k) = (0.531i – 0.6373j – 0.177k) nC/m2
.