For the potential , what is the corresponding electric field at the point (2,2,2)?
E) The correct answer is not given.

Question

For the potential  , what is the corresponding electric field at the point (2,2,2)?
E) The correct answer is not given.

Quizplus · Accepted Answer

The electric field at a point is given by the negative of the gradient of the potential at that point. Therefore, we need to take the gradient of the given potential and evaluate it at the point (2,2,2). This gives us:

E = -grad(V) = -(∂V/∂x i + ∂V/∂y j + ∂V/∂z k)

Note that the coefficients of i, j, and k are the partial derivatives with respect to x, y, and z, respectively. Evaluating these partial derivatives gives:

∂V/∂x = -4x/(x^2 + y^2 + z^2)^(3/2) = -4/(6^(3/2))
∂V/∂y = -4y/(x^2 + y^2 + z^2)^(3/2) = -4/(6^(3/2))
∂V/∂z = -4z/(x^2 + y^2 + z^2)^(3/2) = -4/(6^(3/2))

Therefore, the electric field at the point (2,2,2) is:

E = -∂V/∂x i - ∂V/∂y j - ∂V/∂z k = (4/6^(3/2)) i + (4/6^(3/2)) j + (4/6^(3/2)) k

Simplifying this expression, we get:

E = (2/3^(3/2)) i + (2/3^(3/2)) j + (2/3^(3/2)) k

Which corresponds to choice A.

For the Potential , What Is the Corresponding Electric