Question : An infinite number of charges, each of coulomb, are placed along x-axis at x=1m, 3m, 9m and so on. Calculate the electric field at x=0 due to these charges if all the charges are of the same sign.
Doubt by Deepanshi G.
Solution :
We know,
Electric Field due to a single point charge is given by E=kq/r²
E1=kq/(1)2 = kq/1
E2=kq/(3)2 = kq/9
E3=kq/(9)2 = kq/81
and so on.
Now by using superposition principle
E = E1+E2+E3+...
E= kq+kq/9+kq/81+...
E=kq[1+1/9+1/81+...]
Using the formula for the sum of Infinite GP
Sn=a/(1-r)
Here a=1
r=a2/a1
r=(1/9)/1
r=1/9
So,
E=kq[1/(1-1/9)]
E=kq[1/(8/9)]
E=kq×(9/8)
E=9kq/8
E=(9×9×109)/8 [∵q=1C & k=9×109 Nm2/C2]
E=(81/8)×109
E= 10.125×109
E=1.0125×1010 N/C
