Question

If \(\vec{A} = \vec{\nabla} \times \vec{F}\), then \(\oiint_S \vec{A} \cdot \hat{n} dS\) (for any closed surface S) is:

• A. 0
• B. 4S
• C. 3S
• D. 4V

Hint:

The identity \(\text{div}(\text{curl} \, \vec{F}) = 0\) is extremely useful. It implies that a vector field which is the curl of another field (like the magnetic field \(\vec{B} = \vec{\nabla} \times \vec{A}\)) must be solenoidal (divergence-free). This means it has no sources or sinks, and its field lines must form closed loops.

Answer 1

The Correct Option is A

Step 1: Apply the Divergence Theorem (Gauss's Theorem). The Divergence Theorem relates a surface integral over a closed surface S to a volume integral over the volume V enclosed by that surface: \[ \oiint_S \vec{A} \cdot \hat{n} dS = \iiint_V (\vec{\nabla} \cdot \vec{A}) dV \]
Step 2: Substitute the given expression for \(\vec{A}\). We are given that \(\vec{A} = \vec{\nabla} \times \vec{F}\). Substituting this into the volume integral: \[ \iiint_V \vec{\nabla} \cdot (\vec{\nabla} \times \vec{F}) dV \]
Step 3: Use the vector calculus identity for the divergence of a curl. A fundamental identity in vector calculus is that the divergence of the curl of any vector field is always identically zero: \[ \vec{\nabla} \cdot (\vec{\nabla} \times \vec{F}) = 0 \]
Step 4: Evaluate the integral. Since the integrand is zero, the volume integral is zero. \[ \iiint_V (0) dV = 0 \] Therefore, the surface integral is also zero.