In a region of space, the electric potential is $V(x,y,z)=4xy-7x^2+3yz$. Calculate the $x$, $y$, and $z$ components of the electric field.

• A. $E_x=-4y+14x,E_y=-4x-3z,E_z=-3x$
• B. $E_x=-4y+14x,E_y=-4x-3z,E_z=-3y$
• C. $E_x=-4y-3z,E_y=-4x+14y,E_z=-3y$
• D. $E_x=-4y-3y,E_y=-4x+14y,E_z=-3x$

Answer: (B)

Verified step by step guidance

1. Recall that the electric field \( \vec{E} \) is related to the electric potential \( V(x,y,z) \) by the negative gradient: \[ \vec{E} = -\nabla V = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right) \]
2. Calculate the partial derivative of \( V \) with respect to \( x \): \[ \frac{\partial V}{\partial x} = \frac{\partial}{\partial x} (4xy - 7x^2 + 3yz) = 4y - 14x + 0 = 4y - 14x \]
3. Calculate the partial derivative of \( V \) with respect to \( y \): \[ \frac{\partial V}{\partial y} = \frac{\partial}{\partial y} (4xy - 7x^2 + 3yz) = 4x + 0 + 3z = 4x + 3z \]
4. Calculate the partial derivative of \( V \) with respect to \( z \): \[ \frac{\partial V}{\partial z} = \frac{\partial}{\partial z} (4xy - 7x^2 + 3yz) = 0 + 0 + 3y = 3y \]
5. Combine these results to write the components of the electric field: \[ E_x = -\frac{\partial V}{\partial x} = -(4y - 14x) = -4y + 14x \] \[ E_y = -\frac{\partial V}{\partial y} = -(4x + 3z) = -4x - 3z \] \[ E_z = -\frac{\partial V}{\partial z} = -3y \]