Two long charged thinwalled concentric shells have radii of

Two long, charged thin-walled, concentric shells have radii of 3.7 and 5.9 cm. The charge per unit length is 4.4 x 10_-6 C/m on the inner shell and -7.2 x 10_6 C/m on the outer shell. What are the (a) magnitude E and (b) direction (radially inward or outward) of the electric field at radial distance r = 5.1 cm? What are (c) E and (d) the direction at r = 10 cm? Number Units Number Units

Solution

a) Consider a Cylindrical surface around the inner cylinder concentric with its axis and of radius r = 5.1 cm and of height h

The Cylindrical surface including its top and bottom surface is a closed Gaussian surface.

The total charge enclosed inside the surface Q = 4.4e-6*h C

The cylindrical surface area S = 2h*5.1e-2 sq.m

The electric field E every where on the surface is normal to the surface along the radius and is uniform as the inner cylinder is a long one and has uniform charge distribution

Electric filed through the top and bottom surfaces is 0 as the field is along the radius of the cylinder.

ò_s E.ds = Q/₀

LHS = E*2h*5.1e-2

E*2h*5.1e-2 = 4.4e-6*h/₀

E = 4.4e-6/2*5.1e-2*8.854e-12

     =1.55e+6 N/C

The field is directed radially outward.

Same as above construct a Gaussian surface concentric with the inner and out cylinder having a radius of 10cm and of height h

The Guasian surface includes the charge of the inner cylinder and the outer cylinder.

Total charge enclosed Q = (4.4-7.2)e-6h

Cylindrical surface area = 2h*0.01 sq.m

E is the field on the Gaussian surface

E*2h*0.01 = -2.8*h/₀

E = -2.8e-6/(2*0.01*8.854e-12)

    = -5.56e+5 N/C

-ve sign indicates the field is directed radially inward.