If the net electric field at point P along Y axis is zero, then the ratio of $\left|\frac{\mathrm{q}_{2}}{\mathrm{q}_{3}}\right|$ is $\frac{8}{5 \sqrt{\mathrm{x}}}$, where $\mathrm{x}=$ ______

JEE Main · 2024 · 08 Apr 2024 · Shift 2 · Electrostatics

If the net electric field at point P along Y axis is zero, then the ratio of

\frac{q _{2}}{q _{3}}

\frac{8}{5 x}

, where

x =

______

Answer: 5

q_{2}

is at horizontal distance 2 cm,

q_{3}

at 3 cm, both 4 cm below P. Distances:

r_{2} = 2^{2} + 4^{2} = 20

r_{3} = 3^{2} + 4^{2} = 5

.
The field's y-component from each charge is

\frac{K q}{r ^{2}} \cdot \frac{4}{r} = \frac{4 K q}{r ^{3}}

. For the net y-field to vanish:

\frac{q _{2}}{r _{2}^{3}} = \frac{q _{3}}{r _{3}^{3}} ⟹ \frac{q _{2}}{q _{3}} = \frac{( 20 ) ^{3}}{5 ^{3}} = \frac{20 20}{125} = \frac{8 5}{25} = \frac{8}{5 5}

Comparing with

\frac{8}{5 x}

gives

x = 5

JEE Main · 2024 · 08 Apr 2024 · Shift 2 · Integer answer