Question

If a vector field $\overrightarrow{V}$ is related to another vector field $\overrightarrow{A}$ through $\overrightarrow{V}=\nabla \times \overrightarrow{A}$, which of the following is true? Note: C and $S_C$ refer to any closed contour and any surface whose boundary is C.

• A. $\underset{C}{\oint }\overrightarrow{V}\cdot \overrightarrow{d}l=\underset{S_C}{\int }\overrightarrow{A}\cdot \overrightarrow{dS}$
• B. $\underset{C}{\oint }\overrightarrow{A}\cdot \overrightarrow{d}l=\underset{S_C}{\int }\int \overrightarrow{V}\cdot \overrightarrow{dS}$
• C. $\underset{C}{\oint }\nabla \times \overrightarrow{V}\cdot \overrightarrow{d}l=\underset{S_C}{\int }\int \nabla \times \overrightarrow{A}\cdot \overrightarrow{dS}$
• D. $\underset{C}{\oint }\nabla \times \overrightarrow{A}\cdot \overrightarrow{d}l=\underset{S_C}{\int }\int \overrightarrow{V}\cdot \overrightarrow{dS}$

Answer 1

The correct option is B

$\underset{C}{\oint }\overrightarrow{A}\cdot \overrightarrow{d}l=\underset{S_C}{\int }\int \overrightarrow{V}\cdot \overrightarrow{dS}$

Ans : (b)
By stoke's theorem ,

$\underset{C}{\oint }\overrightarrow{A}\cdot \overrightarrow{d}l=\underset{S_C}{\int }\int \overrightarrow{V}\cdot \overrightarrow{dS}$