Question

The value of \( v_3 \) for which the vector \( \vec{v} = e^y \sin x \hat{i} + e^y \cos x \hat{j} + v_3 \hat{k} \) is solenoidal, is:

• A. \( 2ze^y \cos x \)
• B. \( -2ze^y \cos x \)
• C. \( -2e^y \cos x \)
• D. \( 2e^y \sin x \)

Hint:

Remember the physical meanings of the vector operators:
\textbf{Divergence (\(\nabla \cdot \vec{v}\))}: Measures the rate of expansion from a point (source or sink). Solenoidal (\(\nabla \cdot \vec{v} = 0\)) means the field is incompressible.
\textbf{Curl (\(\nabla \times \vec{v}\))}: Measures the rotation or circulation at a point. Irrotational (\(\nabla \times \vec{v} = \vec{0}\)) means the field is conservative.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A vector field \( \vec{v} \) is called solenoidal if its divergence is zero everywhere. The divergence of a vector field \( \vec{v} = v_1 \hat{i} + v_2 \hat{j} + v_3 \hat{k} \) is given by \( \nabla \cdot \vec{v} = \frac{\partial v_1}{\partial x} + \frac{\partial v_2}{\partial y} + \frac{\partial v_3}{\partial z} \).

Step 2: Key Formula or Approach:
We are given \( v_1 = e^y \sin x \), \( v_2 = e^y \cos x \), and an unknown component \( v_3 \). We will set the divergence of \( \vec{v} \) to zero and solve for \( v_3 \). \[ \nabla \cdot \vec{v} = \frac{\partial}{\partial x}(e^y \sin x) + \frac{\partial}{\partial y}(e^y \cos x) + \frac{\partial v_3}{\partial z} = 0 \]
Step 3: Detailed Explanation:
First, compute the partial derivatives of the known components: \[ \frac{\partial v_1}{\partial x} = \frac{\partial}{\partial x}(e^y \sin x) = e^y \cos x \] \[ \frac{\partial v_2}{\partial y} = \frac{\partial}{\partial y}(e^y \cos x) = e^y \cos x \] Now, substitute these into the divergence equation: \[ (e^y \cos x) + (e^y \cos x) + \frac{\partial v_3}{\partial z} = 0 \] \[ 2e^y \cos x + \frac{\partial v_3}{\partial z} = 0 \] This gives a partial differential equation for \( v_3 \): \[ \frac{\partial v_3}{\partial z} = -2e^y \cos x \] To find \( v_3 \), we integrate this expression with respect to \(z\), treating \(x\) and \(y\) as constants: \[ v_3(x,y,z) = \int (-2e^y \cos x) dz = -2e^y \cos x \int dz = -2ze^y \cos x + C(x,y) \] The term \( C(x,y) \) is an arbitrary function that depends only on \(x\) and \(y\). The options provided are specific functions, implying we should choose the simplest form, which corresponds to setting the integration constant (or function) to zero. Thus, a possible value for \( v_3 \) is \( -2ze^y \cos x \).

Step 4: Final Answer:
The value of \(v_3\) for which the vector is solenoidal is \( -2ze^y \cos x \).