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Find the electric field inside the sphere which carries a charge density proportional to the distance from the origin
ρ = kr
ρ = kr
- a)ρ/ε0
- b)ρr/ε0
- c)ρr2/ε0
- d)none of the above
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Find the electric field inside the sphere which carries a charge densi...
We can start by using Gauss's Law to find the electric field. Gauss's Law states that the flux of the electric field through any closed surface is proportional to the charge enclosed by the surface. Mathematically, it can be written as:
∮E⋅dA = Qenc/ε0
where E is the electric field, dA is an infinitesimal area element on the surface, Qenc is the charge enclosed by the surface, and ε0 is the permittivity of free space.
In this case, we can choose a spherical Gaussian surface centered at the origin, with radius r. The charge enclosed by this surface is:
Qenc = ∫ρdV
where ρ is the charge density and dV is an infinitesimal volume element. Since the charge density is proportional to the distance from the origin, we can write:
ρ = k r
where k is a constant of proportionality. The integral becomes:
Qenc = ∫ρdV = k ∫r^2sinθdrdθdφ
where the limits of integration are 0 to r for r, 0 to π for θ, and 0 to 2π for φ. Evaluating the integral gives:
Qenc = (4/3)πk r^3
Now we can apply Gauss's Law to find the electric field. The flux of the electric field through the Gaussian surface is:
∮E⋅dA = E(4πr^2)
where we have used the fact that the surface area of a sphere is 4πr^2. Therefore, Gauss's Law gives us:
E(4πr^2) = (4/3)πk r^3/ε0
Solving for E, we get:
E = k r/3ε0
Therefore, the electric field inside the sphere is proportional to the distance from the origin, with a constant of proportionality k/3ε0.
∮E⋅dA = Qenc/ε0
where E is the electric field, dA is an infinitesimal area element on the surface, Qenc is the charge enclosed by the surface, and ε0 is the permittivity of free space.
In this case, we can choose a spherical Gaussian surface centered at the origin, with radius r. The charge enclosed by this surface is:
Qenc = ∫ρdV
where ρ is the charge density and dV is an infinitesimal volume element. Since the charge density is proportional to the distance from the origin, we can write:
ρ = k r
where k is a constant of proportionality. The integral becomes:
Qenc = ∫ρdV = k ∫r^2sinθdrdθdφ
where the limits of integration are 0 to r for r, 0 to π for θ, and 0 to 2π for φ. Evaluating the integral gives:
Qenc = (4/3)πk r^3
Now we can apply Gauss's Law to find the electric field. The flux of the electric field through the Gaussian surface is:
∮E⋅dA = E(4πr^2)
where we have used the fact that the surface area of a sphere is 4πr^2. Therefore, Gauss's Law gives us:
E(4πr^2) = (4/3)πk r^3/ε0
Solving for E, we get:
E = k r/3ε0
Therefore, the electric field inside the sphere is proportional to the distance from the origin, with a constant of proportionality k/3ε0.
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