Question 37

The electric potential as a function of $$x, y$$ is given by $$V = 5(x^2 - y^2)$$ V. The electric field at a point $$(2, 3)$$ m is __________ V/m.

$$\vec{E} = -\nabla V = -\left( \frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} \right)$$

$$V = 5x^2 - 5y^2$$

$$E_x = -\frac{\partial V}{\partial x} = -10x$$

$$E_y = -\frac{\partial V}{\partial y} = -(-10y) = 10y$$

$$\vec{E} = -10x\hat{i} + 10y\hat{j}$$

At $$(2, 3)$$: $$\vec{E} = -10(2)\hat{i} + 10(3)\hat{j}$$

$$\vec{E} = -20\hat{i} + 30\hat{j}$$

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