Electric Field from Voltage

One of the values of calculating the scalar electric potential (voltage) is that the electric field can be calculated from it. The component of electric field in any direction is the negative of rate of change of the potential in that direction.

If the differential voltage change is calculated along a direction ds, then it is seen to be equal to the electric field component in that direction times the distance ds. 如果沿方向ds计算微分电压变化，则发现它等于该方向的电场分量乘以距离ds。 The electric field can then be expressed as 电场可以表示为 This is called a partial derivative. 这称为偏导数。

For rectangular coordinates, the components of the electric field are

Express as a gradient. 表示为梯度

Electric Field as Gradient

The expression of electric field in terms of voltage can be expressed in the vector form

This collection of partial derivatives is called the gradient, and is represented by the symbol ∇ . The electric field can then be written

Expressions of the gradient in other coordinate systems are often convenient for taking advantage of the symmetry of a given physical problem.