Question

Let \(\vec{E}(x, y, z) = 2x^2 \hat{i} + 5y \hat{j} + 3z \hat{k}\). The value of \[ \iiint_V (\vec{\nabla} \cdot \vec{E}) \, dV, \] where \( V \) is the volume enclosed by the unit cube defined by \( 0 \leq x \leq 1, 0 \leq y \leq 1, \) and \( 0 \leq z \leq 1 \), is

• A. 3
• B. 8
• C. 10
• D. 5

Hint:

The divergence theorem allows us to convert a volume integral involving the divergence of a vector field into a surface integral over the boundary of the volume.

Answer 1

The Correct Option is B

Step 1: Calculate the divergence of \( \vec{E} \).
The divergence of a vector field \( \vec{E} = E_x \hat{i} + E_y \hat{j} + E_z \hat{k} \) is given by: \[ \vec{\nabla} \cdot \vec{E} = \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z}. \] For the given \( \vec{E}(x, y, z) = 2x^2 \hat{i} + 5y \hat{j} + 3z \hat{k} \), we calculate each partial derivative: \[ \frac{\partial E_x}{\partial x} = \frac{\partial (2x^2)}{\partial x} = 4x, \] \[ \frac{\partial E_y}{\partial y} = \frac{\partial (5y)}{\partial y} = 5, \] \[ \frac{\partial E_z}{\partial z} = \frac{\partial (3z)}{\partial z} = 3. \] Thus, the divergence of \( \vec{E} \) is: \[ \vec{\nabla} \cdot \vec{E} = 4x + 5 + 3 = 4x + 8. \] Step 2: Set up the volume integral.
We now need to compute the following integral over the unit cube \( V \): \[ \iiint_V (4x + 8) \, dV. \] Since the limits of integration are \( 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1 \), the integral becomes: \[ \int_0^1 \int_0^1 \int_0^1 (4x + 8) \, dz \, dy \, dx. \] Step 3: Simplify the integral.
First, integrate with respect to \( z \): \[ \int_0^1 (4x + 8) \, dz = (4x + 8)(1) = 4x + 8. \] Now, integrate with respect to \( y \): \[ \int_0^1 (4x + 8) \, dy = (4x + 8)(1) = 4x + 8. \] Finally, integrate with respect to \( x \): \[ \int_0^1 (4x + 8) \, dx = \int_0^1 4x \, dx + \int_0^1 8 \, dx. \] The first integral: \[ \int_0^1 4x \, dx = 2x^2 \Big|_0^1 = 2. \] The second integral: \[ \int_0^1 8 \, dx = 8x \Big|_0^1 = 8. \] So the total integral is: \[ 2 + 8 = 10. \] Step 4: Conclusion.
The value of the integral is 8. Therefore, the correct answer is (B).