Q. Two small charged spheres $A$ and $B$ have charges $ 10\,\mu \,C $ and $ 40\,\mu \,C $ , respectively and are held at a separation of $90\, cm$ from each other At what distance from $A$, electric intensity would be zero?
Delhi UMET/DPMTDelhi UMET/DPMT 2005
Solution:
Let a test charge $q_{0}$ is placed at a point $D$ at a distance $x$ from point $A$.
From Coulombs law, the force on the test charge $q_{0}$ due to a charge $q$, kept a distance $r$ apart is
$F=-\frac{1}{4 \pi \varepsilon_{0}} \frac{q q_{0}}{r^{2}}$
$\therefore $ Electric field intensity at the point $D$ $E=\frac{F}{4 \pi \varepsilon_{0}}=\frac{1}{4 \pi \varepsilon_{0}} \frac{q q_{0}}{r^{2}}$
Let intensity at $D$ distance $x$ from point A be zero,
then $\frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{10 \times 10^{-6}}{x^{2}}$
$=\frac{1}{4 \pi \varepsilon_{0}} \frac{40 \times 10^{-6}}{(90-x)^{2}}$ $\Rightarrow \frac{1}{x^{2}}=\frac{4}{(90-x)^{2}}$
$\Rightarrow \frac{1}{x}=\frac{2}{90-x}$
$\Rightarrow $ $90-x=2 x$
$\Rightarrow 3 x=90$
$\Rightarrow x=30\, cm$.
Note: The force $F$ acts along the line joining the two charges.
