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The electirc potential at a point (x, y,...

The electirc potential at a point `(x, y, z)` is given by
`V = -x^(2)y - xz^(3) + 4`
The electric field `vecE` at that point is

A

`E = (2xy +z^(3))hati + x^(2)hatj +3xz^(2) hatk`

B

`E = 2xy hati +(x^(2) +y^(2)) hatj + (3xz - y^(2))hatk`

C

`E = z^(3) hati +xyzhatj +z^(2) hatk`

D

`E = (2xy - z^(3)) hayi +xy^(2) hatj +3z^(2) xx hatk`

Text Solution

AI Generated Solution

The correct Answer is:
To find the electric field \(\vec{E}\) at the point \((x, y, z)\) given the electric potential \(V\), we can use the relationship between electric potential and electric field: \[ \vec{E} = -\nabla V \] Where \(\nabla V\) is the gradient of the potential \(V\). The gradient in three dimensions is given by: \[ \nabla V = \left( \frac{\partial V}{\partial x}, \frac{\partial V}{\partial y}, \frac{\partial V}{\partial z} \right) \] ### Step 1: Find the partial derivative of \(V\) with respect to \(x\) Given the potential: \[ V = -x^2 y - x z^3 + 4 \] We differentiate \(V\) with respect to \(x\): \[ \frac{\partial V}{\partial x} = -\frac{\partial}{\partial x}(x^2 y) - \frac{\partial}{\partial x}(x z^3) \] Calculating these derivatives: \[ \frac{\partial V}{\partial x} = -2xy - z^3 \] ### Step 2: Find the partial derivative of \(V\) with respect to \(y\) Next, we differentiate \(V\) with respect to \(y\): \[ \frac{\partial V}{\partial y} = -\frac{\partial}{\partial y}(x^2 y) = -x^2 \] ### Step 3: Find the partial derivative of \(V\) with respect to \(z\) Now, we differentiate \(V\) with respect to \(z\): \[ \frac{\partial V}{\partial z} = -\frac{\partial}{\partial z}(x z^3) = -3xz^2 \] ### Step 4: Combine the results to find \(\vec{E}\) Now we can combine these results to find the electric field \(\vec{E}\): \[ \vec{E} = -\nabla V = -\left( \frac{\partial V}{\partial x}, \frac{\partial V}{\partial y}, \frac{\partial V}{\partial z} \right) \] Substituting the derivatives we found: \[ \vec{E} = -\left( -2xy - z^3, -x^2, -3xz^2 \right) \] This simplifies to: \[ \vec{E} = (2xy + z^3) \hat{i} + x^2 \hat{j} + 3xz^2 \hat{k} \] ### Final Answer Thus, the electric field \(\vec{E}\) at the point \((x, y, z)\) is: \[ \vec{E} = (2xy + z^3) \hat{i} + x^2 \hat{j} + 3xz^2 \hat{k} \]
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