Question

Prove that electric field at a point is equal to the negative gradient of electrostatic potential at that point

Answer 1

Answer:

The approach is to imagine moving a charge from point A to Point B

Where there is a potential difference between the two points. (Easiest to imagine in an electrostatic field- but it need not be the case)

If you move a positive charge Q through this potential difference, the work done =Q x delta V where delta V is the change in potential/PD.

You can also calculate the work done by using force x distance. If we choose a case where the electric field is uniform, like between parallel plates, this makes the maths easier.

Force on charge = QE (E =elec field strength)

Work done = force x distance so

work done = -QEd or -QE(delta x) where delta x is change in position)

The minus sign is present because the Charge Q is being pushed to the higher potential so that work is being done on the charge. This means that the delta x is in the opposite direction to the movement. If we call the movement in the positive x direction then the electric field is in the opposite direction so it is negative,

Now we can eqaute the two methods for calculating work done

-QE (delta x) = Q delta V (Q can cancel)

-E (delta x) = delta V

E==delta V/ delta x ie minus the potential gradient.

Explanation:

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