Question

1) A thin ring made of uniformly charged insulating material has total charge Q and radius R. The ring is positioned along the x-y plane of a 3d coordinate system such that the center of the ring is at the origin of the coordinate system. (a) Determine an expression for the potential at an arbitrary location along the z-axis in terms of Q, R, and z. (b) Use this expression to determine an expression for the magnitude of the electric field at an arbitrary location along the z-axis in terms of Q, R, and z. Hint: Apply the technique of charge integration in part (a) and poten

Answer 1

(A) considering the charge "q" evenly distributed, applying the technique of charge integration for finite charges, you obtain the expression for the potential along any point in the Z-axis:

`V(z)=\frac{Q}{4\pi (\epsilon_(0)) \sqrt{R^(2) +z^(2)}  }`

With
`(\epsilon_(0))` been the vacuum permittivity

(B) The expression for the magnitude of the E(z) electric field along the Z-axis is:

`E(z)=\frac{QZ}{4\pi (\epsilon_(0)) (R^(2) +z^(2))^{(3)/(2) }    }`

Step-by-step explanation:

(A) Considering a uniform linear density
`λ_(0)` on the ring, then:

`dQ=\lambda dl` (1)⇒
`Q=\lambda_(0) 2\pi R`(2)⇒
`\lambda_(0)=(Q)/(2\pi R)`(3)

Applying the technique of charge integration for finite charges:

`V(z)= 4\pi (ε_(0))\int\limits^a_b {(1)/( r'  )} \, dQ`(4)

Been r' the distance between the charge and the observation point and a, b limits of integration of the charge. In this case a=2π and b=0.

Using cylindrical coordinates, the distance between a point of the Z-axis and a point of a ring with R radius is:

`r'=\sqrt{R^(2) +Z^(2)}`(5)

Using the expressions (1),(4) and (5) you obtain:

`V(z)= 4\pi (\epsilon_(0))\int\limits^a_b {\frac{\lambda_(0)R}{ \sqrt{R^(2) +Z^(2)}  }} \, d\phi`

Integrating results:

`V(z)=\frac{Q}{4\pi (\epsilon_(0)) \sqrt{R^(2) +z^(2)}  }` (S_a)

(B) For the expression of the magnitude of the field E(z), is important to remember:

`|E| =-\\abla V` (6)

But in this case you only work in the z variable, soo the expression (6) can be rewritten as:

`|E| =-(dV(z))/(dz)` (7)

Using expression (7) and (S_a), you get the expression of the magnitude of the field E(z):

`E(z)=\frac{QZ}{4\pi (\epsilon_(0)) (R^(2) +z^(2))^{(3)/(2) }    }` (S_b)