Question

If \( \phi(x,y,z) \) is a scalar function which satisfies the Laplace equation, then the gradient of \( \phi \) is

• A. Solenoidal and irrotational
• B. Solenoidal but not irrotational
• C. Irrotational but not solenoidal
• D. Neither solenoidal nor irrotational

Hint:

The gradient of any scalar field is always irrotational, but not solenoidal.

Answer 1

The Correct Option is A

Step 1: Understanding the Laplace equation.
The Laplace equation is given by: \[ \nabla^2 \phi = 0 \] This equation describes a scalar field where the divergence of the gradient of \( \phi \) is zero.
Step 2: Gradient of a scalar field.
The gradient \( \nabla \phi \) is a vector field. It is always irrotational, meaning its curl is zero: \[ \nabla \times (\nabla \phi) = 0 \] This implies the gradient is irrotational.
Step 3: Solenoidal property.
The gradient of a scalar field is not solenoidal, as the divergence of the gradient of a scalar field is non-zero. In this case, the divergence of \( \nabla \phi \) is zero, so it is not solenoidal.
Step 4: Conclusion.
The gradient of \( \phi \) is irrotational but not solenoidal, so the correct answer is (C).