Question

10. Find the electric field at the origin for a line of charge density λ on the y 0 portion of the unit circle in the r-y plane if (a) λ is constant (b) λ= cos φ

Answer 1

Find the electric field E^rightarrow at the origin for a line of charge density lambda on the y > 0 Now let�s consider a small element charge dw. we denote the electric field produced by it as dE^rightarrow It will have two components x and y components Now the x-component will get cancelled due to symmetry of the charge distribution y component will be there therefore dE_y = 1/4 pi E_0 dw/a^2 sin Q Now, if lambda is the charge density, then + a total charge = (pi a) times lambda = pi a lambda = Q (say) So, Q is distributed over an angle of pi \r\n therefore dw/d emptyset = Q/pi therefore dw = Q/pi d emptyset therefore dE_y = 1/4 pi + 0 1/a^2 Q/pi d emptyset sin Q Now, if + a circle has unit radius i.e a = 1 then dE_y = 1/4 pi E_0 pi lambda/pi sin Q d emptyset therefore dE_y = 1/4 pi E_0 lambda sin Q d emptyset