Electric Field due to Charged Circular

 Electric Field due to Charged Circular

AU : Dec.-14,19

• Consider a charged circular ring of radius r placed in xy plane with centre at origin, carrying a charge uniformly along its circumference. The charge density is ρ_L C/m .

• The point P is at a perpendicular distance 'z' from the ring as shown in the Fig. 2.7.1.

• Consider a small differential length dZ on this ring.

The charge on it is dQ.

where R = Distance of point P from dl

• Consider the cylindrical coordinate system. For dl we are moving in direction where dl = r dϕ

          dl = r dϕ      ... (2.7.2)

Now  R² = r² + z²  ... from Fig. 2.7.1

• While  can be obtained from its two components, in cylindrical system as shown in the Fig. 2.7.1 (a). The two components are,

1. Distance r in the direction of , radially inwards i.e. 

2. Distance z in the direction of 

 ... (2.7.3)

Key Point : This method can be used conveniently to obtain  by identifying its components in the direction of unit vectors in the co-ordinate system considered.

Note : The radial components of  at point P will be symmetrically placed in the plane parallel to xy plane and are going to cancel each other. This is shown in the Fig. 2.7.1 (b).

where r = Radius of the ring

z = Perpendicular distance of point P from the ring along the axis of the ring

• This is the electric field at a point P (0, 0, z) due to the circular ring of radius r placed in xy plane.

Ex. 2.7.1 A circular ring of charge with radius 5 m lies in z = 0 plane with centre at origin. If the ρ_L = 10 nC/m, find the point charge Q placed at the origin which will produce same  at the point (0, 0, 5) m.

Sol. : Use the procedure discussed above giving,

Example for Practice

Ex. 2.7.2 A ring of radius 6 m is placed in yz plane. It is centered at origin. Find electric field intensity at point (8, 0, 0) m. The line charge density is 18 nC/m. 

Review Question

1. Obtain an expression for an electric field due to charged circular ring of radius 'h' placed in xy plane, at a point P (0, 0, z), having uniform line charge density of ρ_L C/m. 

AU : Dec.-14,19, Marks 8