Constant electric field in the sphere

[Tanya Sharma]

Homework Statement

The region between two concentric spheres of radii a and b(>a) contains volume charge density ρ(r) = C / r where C is a constant and r is the radial distance, as shown in figure. A point charge q is placed at the origin, r= 0. Find the value of C for which the electric field in the region between the spheres is constant (i.e., r independent).

A) q/πa²
B) q/π(a²+b²)
C) q/2πa²
D) q/2πb²

Homework Equations

ε∫E.dS = q_free

The Attempt at a Solution

Let us consider a a gaussian surface, a concentric sphere at radius r .

Applying Gauss law

ε∫E.dS = q + 4πC(r²-a²)

εE(4πr²) = q + 4πC(r²-a²)

E = q/(4πr²ε) + C(r²-a²)/(εr²)

Now for E to be constant ,

q-4πca² = 0 or C=q/(4πa²) .This is not given as one of the options .

I would be grateful if somebody could help me with the problem.

 

[SammyS]

Homework Statement

The region between two concentric spheres of radii a and b(>a) contains volume charge density ρ(r) = C / r where C is a constant and r is the radial distance, as shown in figure. A point charge q is placed at the origin, r= 0. Find the value of C for which the electric field in the region between the spheres is constant (i.e., r independent).

A) q/πa²
B) q/π(a²+b²)
C) q/2πa²
D) q/2πb²

Homework Equations

ε∫E.dS = q_free

The Attempt at a Solution

Let us consider a a gaussian surface, a concentric sphere at radius r .

Applying Gauss law

ε∫E.dS = q + 4πC(r²-a²)

εE(4πr²) = q + 4πC(r²-a²)

E = q/(4πr²ε) + C(r²-a²)/(εr²)

Now for E to be constant ,

q-4πca² = 0 or C=q/(4πa²) .This is not given as one of the options .

I would be grateful if somebody could help me with the problem.

You have not integrated the charge correctly.

 

[Tanya Sharma]

Hi SammyS...

Yes.I missed a factor of 2 while integrating .

Does that mean option C) i.e C=q/2πa² is the correct answer ?

 

[SammyS]

Hi SammyS...

Yes.I missed a factor of 2 while integrating .

Does that mean option C) i.e C=q/2πa² is the correct answer ?

That's what I got .

 

[Tanya Sharma]

Thank you very much