Question

$\vec{E} = x\hat{i} + y\hat{j} + z\hat{k}$ exists in space then choose INCORRECT statement(s):

• A. This electric field is not possible
• B. Volume charge density is non-uniform.
• C. If a charge particle goes from (3, 4, 5) to (5, 3, 4) then work done by electric force on charge will be positive
• D. Charge enclosed in sphere of unit radius and centered at origin is $4\pi\epsilon_0$.

Answer 1

Answer: (D)

(A), (B), and (C)

Answer 2

Solution

Given the electric field

$$\vec{E}=x\,\hat{i}+y\,\hat{j}+z\,\hat{k},$$

we check each statement.

1. Statement (A):
   The field is conservative since

$$\nabla\times\vec{E}=\vec{0}.$$

Its potential is

$$V=-\frac{1}{2}(x^2+y^2+z^2)+\text{constant}.$$

Even though the potential diverges at infinity, a field defined everywhere (with appropriate charge distribution) is mathematically acceptable. Thus, saying “this electric field is not possible” is incorrect.

2. Statement (B):
   Using Gauss’s law in differential form,

$$\nabla\cdot\vec{E}=\frac{\partial x}{\partial x}+\frac{\partial y}{\partial y}+\frac{\partial z}{\partial z}=1+1+1=3.$$

Hence, the volume charge density is

$$\rho=\epsilon_0\,(\nabla\cdot\vec{E})=3\epsilon_0,$$

which is uniform. The statement “volume charge density is non-uniform” is incorrect.

3. Statement (C):
   The potential corresponding to $\vec{E}$ is

$$V=-\frac{1}{2}(x^2+y^2+z^2).$$

For points $(3,4,5)$ and $(5,3,4)$,

$$V(3,4,5)=-\frac{1}{2}(9+16+25)=-\frac{50}{2}=-25,$$ $$V(5,3,4)=-\frac{1}{2}(25+9+16)=-\frac{50}{2}=-25.$$

The potential difference is zero, so the work done by the electric force on a charge is zero, not positive. Thus, statement (C) is incorrect.

4. Statement (D):
   For a sphere of unit radius, the flux is

$$\Phi_E=\oint \vec{E}\cdot d\vec{A}=\text{(Field magnitude=1, area }=4\pi)=4\pi.$$

By Gauss’s law,

$$\Phi_E=\frac{Q}{\epsilon_0}\implies Q=4\pi\epsilon_0.$$

Thus, statement (D) is correct.

Answer: The incorrect statements are (A), (B), and (C).

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Explanation (Minimal Core Steps):

1. Calculate divergence: $\nabla\cdot\vec{E}=3$ $\rightarrow$ $\rho=3\epsilon_0$ (uniform).
2. Potential: $V=-\frac{1}{2}(x^2+y^2+z^2)$ $\rightarrow$ same at both points $(3,4,5)$ and $(5,3,4)$ $\rightarrow$ zero work.
3. Flux through a sphere of radius 1: $4\pi$ $\rightarrow$ $Q=4\pi\epsilon_0$.