Question 675: Electrostatics & Capacitance

Question

An infinite number of point charges, each carrying 1 $$\mu$$C charge, are placed along the y-axis at y = 1 m, 2 m, 4 m, 8 m ...............
The total force on a 1C point charge, placed at the origin, is x $$	imes$$ 10
3
N.
The value of x, to the nearest integer, is __________. [Take $${1 \over {4\pi {\varepsilon _0}}} = 9 	imes {10^9}$$ Nm
2
/C
2
]

Accepted Answer

$${F_{total}} = {{k{q_1}{q_2}} \over {r_1^2}} + {{k{q_1}{q_3}} \over {r_2^2}} + {{k{q_1}{q_4}} \over {r_3^2}} + .....$$
$$ = 9 	imes {10^9} 	imes {10^{ - 6}}\left[ {1 + {{\left( {{1 \over 2}} ight)}^2} + {{\left( {{1 \over {{2^2}}}} ight)}^2} + {{\left( {{1 \over {{2^3}}}} ight)}^2} + {{\left( {{1 \over {{2^\infty }}}} ight)}^2}} ight]$$
$$ = 9 	imes {10^9} 	imes {10^{ - 6}}\left[ {{1 \over {1 - {1 \over 4}}}} ight]$$
$$ \because $$ $$\left[ {{S_\infty } = {a \over {1 - r}}} ight]$$ for G.P.
$$ = 9 	imes {10^3} 	imes {4 \over 3}$$ = 12 $$	imes$$ 10
3
N

Electrostatics & Capacitance

Integer / Numeric Answer Type