Potential of a Charged Conducting Sphere

[Estefania_8]

Homework Statement

A conducting sphere with radius R is charged to voltage V0 (relative to a point an infinite distance from the sphere where the potential is zero). What is the surface charge density σ? Express your answer in terms of the given quantities and ϵ0.

Homework Equations

Electric field on a sphere, E=(1/4πϵ0)*(Q/r^2)
Surface charge density: σ=Q/A, where Q is the charge and A is the area of the surface.
A for a sphere: A=4πr^2

The Attempt at a Solution

I need to find σ and I have Q in two of the equations I replaced A with the area of a sphere, so I get
σ=Q/4πr^2
Then I solved for Q from this, so I have Q=σ4πr^2, now I substitute this into the first equation and solve for σ, so now I have, E=(1/4πϵ0r^2)*(σ4πr^2/r^2), this gives me E*ϵ0, but this is not the correct answers. Does E=V0?

 

[PWiz]

What do you mean by 0 here? Are you referring to the electric permittivity of free space, $ε_0$ ?

 

[ehild]

Hi Estefania, welcome to PF!

Homework Statement

A conducting sphere with radius R is charged to voltage V0 (relative to a point an infinite distance from the sphere where the potential is zero). What is the surface charge density σ? Express your answer in terms of the given quantities and ϵ0.

Homework Equations

Electric field on a sphere, E=(1/4πϵ0)*(Q/r^2)
Surface charge density: σ=Q/A, where Q is the charge and A is the area of the surface.
A for a sphere: A=4πr^2

The Attempt at a Solution

I need to find σ and I have Q in two of the equations I replaced A with the area of a sphere, so I get
σ=Q/4πr^2
Then I solved for Q from this, so I have Q=σ4πr^2, now I substitute this into the first equation and solve for σ, so now I have, E=(1/4πϵ0r^2)*(σ4πr^2/r^2), this gives me E*ϵ0, but this is not the correct answers. Does E=V0?

You need to give σ in terms of V₀ , the potential of the sphere. What is the formula for the potential of a charged sphere ?

 

[Estefania_8]

What do you mean by 0 here? Are you referring to the electric permittivity of free space, $ε_0$ ?

Yes, that's what I meant.

 

[Estefania_8]

Hi Estefania, welcome to PF!

You need to give σ in terms of V₀ , the potential of the sphere. What is the formula for the potential of a charged sphere ?

The potential is given by V=Q/4pi*epsilon naught*r, so I would use this equation and set V0 equal to this?

 

[ehild]

The potential is given by V=Q/4pi*epsilon naught*r, so I would use this equation and set V0 equal to this?

Yes, $V_0=\frac{Q}{4\pi\epsilon_0 r}$ . (Use parentheses!)
Find Q from this.

 

[PWiz]

Yes, that's what I meant.

Then you have all the variables you'll need. Follow ehild's procedure and eliminate $Q$ from the equations. This should be easy as you've already stated yourself that $σ=\frac Q{4πr^2}$.

 

[Estefania_8]

Yes, $V_0=\frac{Q}{4\pi\epsilon_0 r}$ . (Use parentheses!)
Find Q from this.

Yes, $V_0=\frac{Q}{4\pi\epsilon_0 r}$ . (Use parentheses!)
Find Q from this.

OK, so now I have V0*4πε0*r=Q, so now I would just use σ=Q/A, which would translate into σ=Q/4πr^2, like PWiz said. Then I would use σ4πr^2=Q, which would give me V0*4πε0*r=σ4πr^2, so solving for σ I get Voε0/r=σ and that is the answer. Thank you to you both!

 

[ehild]

You are welcome