Explanation:

Definition of Electric Field Intensity and Electric Potential

- The electric field intensity (E) is a vector quantity that represents the force per unit charge experienced by a test charge at a given point in space.
- The electric potential (V) is a scalar quantity that represents the electric potential energy per unit charge of a test charge at a given point in space.

Relationship between Electric Field Intensity and Electric Potential

- The electric field intensity is the negative gradient of the electric potential. This means that:

E = -∇V

- Here, ∇ is the gradient operator, which represents the rate of change of a function with respect to its spatial coordinates.
- The negative sign indicates that the electric field points in the direction of decreasing potential.

Proof of the Relationship

- The proof of this relationship can be done using the definition of the gradient and the electric potential.
- The gradient of the electric potential is given by:

∇V = (∂V/∂x)i + (∂V/∂y)j + (∂V/∂z)k

- Here, i, j, and k are the unit vectors along the x, y, and z axes, respectively.
- The electric field intensity is given by:

E = F/q

- Here, F is the force experienced by a test charge q in the electric field.
- Using the definition of electric potential energy, we can write:

F = -∇Vq

- Substituting this in the expression for the electric field intensity, we get:

E = -∇V

- This proves the relationship between the electric field intensity and the electric potential.

Conclusion

- Therefore, the statement "The electric field intensity is the negative gradient of the electric potential" is true.