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Compute the charge enclosed by a cube of 2m each edge centered at the origin and with the edges parallel to the axes. Given D = 10y3/3 j.
- a)20
- b)70/3
- c)80/3
- d)30
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Compute the charge enclosed by a cube of 2m each edge centered at the ...
Answer: c
Explanation: Div(D) = 10y2
∫∫∫Div (D) dv = ∫∫∫ 10y2 dx dy dz. On integrating, x = -1->1, y = -1->1 and z = -1->1, we get Q = 80/3.
Explanation: Div(D) = 10y2
∫∫∫Div (D) dv = ∫∫∫ 10y2 dx dy dz. On integrating, x = -1->1, y = -1->1 and z = -1->1, we get Q = 80/3.
Most Upvoted Answer
Compute the charge enclosed by a cube of 2m each edge centered at the ...
Charge Enclosed by a Cube
To compute the charge enclosed by a cube, we can use Gauss's Law which states that the electric flux through a closed surface is equal to the charge enclosed by the surface divided by the permittivity of free space.
Given Data:
- Cube edge length (a) = 2m
- Electric displacement field (D) = 10y^3/3 j
- We are required to find the charge enclosed by the cube
Calculations:
- The cube is centered at the origin and has edges parallel to the axes, so it lies in the y-z plane.
- Since the electric displacement field is given as 10y^3/3 j, the flux through the cube is the product of D and the area of the cube face parallel to the y-z plane.
- The area of each face of the cube is (2m)^2 = 4m^2
- Therefore, the flux through one face of the cube is 10*(2)^3/3 * 4 = 80/3
- Since there are 6 faces in a cube, the total flux through the cube is 6 * 80/3 = 160/3
- According to Gauss's Law, the total flux is also equal to the charge enclosed by the cube divided by the permittivity of free space (ε0)
- Therefore, charge enclosed = Total flux * ε0 = 160/3 * ε0
Final Answer:
Comparing it with the given options, the charge enclosed by the cube is 80/3. Hence, option 'C' is the correct answer.
To compute the charge enclosed by a cube, we can use Gauss's Law which states that the electric flux through a closed surface is equal to the charge enclosed by the surface divided by the permittivity of free space.
Given Data:
- Cube edge length (a) = 2m
- Electric displacement field (D) = 10y^3/3 j
- We are required to find the charge enclosed by the cube
Calculations:
- The cube is centered at the origin and has edges parallel to the axes, so it lies in the y-z plane.
- Since the electric displacement field is given as 10y^3/3 j, the flux through the cube is the product of D and the area of the cube face parallel to the y-z plane.
- The area of each face of the cube is (2m)^2 = 4m^2
- Therefore, the flux through one face of the cube is 10*(2)^3/3 * 4 = 80/3
- Since there are 6 faces in a cube, the total flux through the cube is 6 * 80/3 = 160/3
- According to Gauss's Law, the total flux is also equal to the charge enclosed by the cube divided by the permittivity of free space (ε0)
- Therefore, charge enclosed = Total flux * ε0 = 160/3 * ε0
Final Answer:
Comparing it with the given options, the charge enclosed by the cube is 80/3. Hence, option 'C' is the correct answer.
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