Section 2.5: Field Along Axis of Symmetry
In highly symmetric charge distributions (such as rings, disks, or spheres), the electric field along the axis of symmetry can be calculated using integration and symmetry arguments.
For a uniformly charged ring of radius \( R \) and total charge \( Q \), the electric field at a point on the axis a distance \( x \) from the center is
\[ E = \frac{1}{4 \pi \epsilon_0} \cdot \frac{Qx}{(x^2 + R^2)^{3/2}} \]
This is directed along the axis of the ring.
Example 1
Find the field on the axis of a uniformly charged disk of radius \( R \) and surface charge density \( \sigma \).
Consider the disk as concentric rings of radius \( r \) and width \( dr \).
Charge on ring: \( dq = \sigma \cdot 2 \pi r \, dr \).
Field contribution: \[ dE = \frac{1}{4 \pi \epsilon_0} \cdot \frac{dq \cdot x}{(x^2 + r^2)^{3/2}} \]
Integrate from \( 0 \) to \( R \): \[ E = \frac{\sigma}{2 \epsilon_0} \left( 1 - \frac{x}{\sqrt{x^2 + R^2}} \right) \]
This is directed perpendicular to the disk surface (along the axis).
Practice Problems
- Derive the field of a uniformly charged thin ring at a point on its axis.
- Show that far from a charged disk (\( x \gg R \)), the field reduces to that of a point charge.
- Find the field on the axis of a uniformly charged finite rod.
- Conceptual: Why does symmetry eliminate all field components except along the axis?
- Compute the field at the center of a square sheet of charge using symmetry arguments.