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A quick question that is currently bothering me.

I have the following equation:

$\mathbf{E}+\frac{\partial \mathbf{A}}{\partial t} = -\nabla V$

My question is, how can the right side, being a vector, ever become a scalar ?

Or am I missing something fundamental here ? (The equation is from my Electrodynamics book)

Thanks in advance.

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2 Answers 2

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The gradient of a scalar is again vector.

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Both sides of the equation given are vectors and so represent 3 equations which are, on a Cartesian basis:

$E_x + \frac{\partial A_x}{\partial t} = -\frac{\partial V}{\partial x} $

$E_y + \frac{\partial A_y}{\partial t} = -\frac{\partial V}{\partial y} $

$E_z + \frac{\partial A_z}{\partial t} = -\frac{\partial V}{\partial z} $

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