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Step-by-Step Solution
Step 1: Identify the Relevant Concept (Gauss’s Law)
Gauss’s law states that the total electric flux through a closed surface is equal to the net charge enclosed by that surface, divided by \epsilon_0 . Mathematically:
\Phi_E = \frac{Q}{\epsilon_0}.
Step 2: Note the Given Electric Field
The electric field in the region is given by:
\overrightarrow{E} = E_0 \, x \, \hat{i},
where E_0 = 4 \times 10^4 \,\mathrm{N\,C^{-1}\,m^{-1}} and x is the x -coordinate in meters.
Step 3: Dimensions of the Cube
The cube is bounded by x = 0, x = a, y = 0, y = a, z = 0, z = a , where
a = 2\,\mathrm{cm} = 0.02\,\mathrm{m}.
Step 4: Compute the Electric Flux Through the Cube
We calculate the flux through each face of the cube. The total electric flux is:
\Phi_E = \oint \overrightarrow{E} \cdot d\overrightarrow{A}.
Face at x = 0 :
Here, x=0 , so \overrightarrow{E} = E_0 \times 0 \times \hat{i} = 0 .
Hence, the flux through this face is zero.
Face at x = a :
Here, x=a , so \overrightarrow{E} = E_0 \, a \, \hat{i}.
Substitute:
a = 0.02\,\mathrm{m}, \quad E_0 = 4 \times 10^4 \,\mathrm{N\,C^{-1}\,m^{-1}}.
E = (4 \times 10^4)\times 0.02 = 800\,\mathrm{N\,C^{-1}}.
The area of this face is:
A = a^2 = (0.02)^2 = 4 \times 10^{-4}\,\mathrm{m^2}.
Hence, the flux through this face is:
\Phi_{x=a} = E \times A = 800 \times 4 \times 10^{-4} = 0.32\,\mathrm{N\,m^2\,C^{-1}}.
Faces at y=0, y=a and z=0, z=a :
The field \overrightarrow{E} is along \hat{i} , so it is parallel to these faces. Hence, \overrightarrow{E} \cdot d\overrightarrow{A} = 0 for these faces, yielding zero flux through them.
Therefore, the total electric flux is
\Phi_E = 0 + 0.32 + 0 = 0.32\,\mathrm{N\,m^2\,C^{-1}}.
Step 5: Use Gauss’s Law to Determine Enclosed Charge
By Gauss’s law, Q = \Phi_E \, \epsilon_0 . Substituting
\epsilon_0 = 9 \times 10^{-12}\,\mathrm{C^2\,N^{-1}\,m^{-2}},
Q = 0.32 \times (9 \times 10^{-12}) = 2.88 \times 10^{-12}\,\mathrm{C}.
Step 6: Convert to the Required Form
The problem asks for the enclosed charge as Q \times 10^{-14}\,\mathrm{C} . Rewrite:
2.88 \times 10^{-12}\,\mathrm{C}
= 2.88 \times 10^{-12} \times \frac{10^{14}}{10^{14}} \,\mathrm{C}
= (2.88 \times 10^2) \times 10^{-14}\,\mathrm{C}
= 288 \times 10^{-14}\,\mathrm{C}.
Therefore, Q = 288.
Final Answer
The value of Q (when expressed as Q \times 10^{-14}\,\mathrm{C} ) is 288.
