- #1
doktorwho
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1. Homework Statement
A flat ring, with radii ##b>a## like on the picture is charged with a surface charge density ##ρ_s=ρ_0\frac{b}{r}## where ##ρ_0=const.## Calculate the potential of the ring on the ##z-axis## by taking the reference point to be in the infinity. Picture below.
2. Homework Equations
Since the reference point is in the infinty the expression for the potential is $$dV=\frac{dQ}{4πε_0r}$$
I have some ideas and have tried to solve the problem but need the confirmation if this is correct or not.
It goes like this.
1) First i calculated the expression for the total charge on the ring: note: (##ρ_l## is the linear charge density on one circle of the ring)
##ρ_l=ρ_sdA=ρ_s*rdθ=ρ_s2rπ##
##dQ=ρ_s2rπdr##
2)Then i wrote the expression for the potential: note: ( ##R## is the distance to some point on the ##z-axis##
##dV=\frac{dQ}{4πε_0R}##
##V=\int_{a}^{b}\frac{ρ_srdr}{2ε_0\sqrt{z^2+r^2}}##
What do you think about this? Is it correct? Is there an easier way?[/B]
A flat ring, with radii ##b>a## like on the picture is charged with a surface charge density ##ρ_s=ρ_0\frac{b}{r}## where ##ρ_0=const.## Calculate the potential of the ring on the ##z-axis## by taking the reference point to be in the infinity. Picture below.
2. Homework Equations
Since the reference point is in the infinty the expression for the potential is $$dV=\frac{dQ}{4πε_0r}$$
The Attempt at a Solution
I have some ideas and have tried to solve the problem but need the confirmation if this is correct or not.
It goes like this.
1) First i calculated the expression for the total charge on the ring: note: (##ρ_l## is the linear charge density on one circle of the ring)
##ρ_l=ρ_sdA=ρ_s*rdθ=ρ_s2rπ##
##dQ=ρ_s2rπdr##
2)Then i wrote the expression for the potential: note: ( ##R## is the distance to some point on the ##z-axis##
##dV=\frac{dQ}{4πε_0R}##
##V=\int_{a}^{b}\frac{ρ_srdr}{2ε_0\sqrt{z^2+r^2}}##
What do you think about this? Is it correct? Is there an easier way?[/B]
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