An infinite plane sheet of charge having uniform surface charge density \(+\sigma_{\mathrm{s}} \mathrm{C} / \mathrm{m}^{2}\) is placed on \(\mathrm{x}-\mathrm{y}\) plane. Another infinitely long line charge having uniform linear charge density \(+\lambda_{\mathrm{e}} \mathrm{C} / \mathrm{m}\) is placed at \(\mathrm{z}=4 \mathrm{~m}\) plane and parallel to y -axis. If the magnitude values \(\left|\sigma_{\mathrm{s}}\right|=2\left|\lambda_{\mathrm{e}}\right|\) then at point ( \(0,0,2\) ), the ratio of magnitudes of electric field values due to sheet charge to that of line charge is \(\pi \sqrt{n}: 1\). The value of \(n\) is \_\_\_\_ .

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Electrostatics

An infinite plane sheet of charge having uniform surface charge density

+ σ_{s} C / m^{2}

is placed on

x - y

plane. Another infinitely long line charge having uniform linear charge density

+ λ_{e} C / m

is placed at

z = 4 m

plane and parallel to y -axis. If the magnitude values

∣ σ_{s} ∣ = 2 ∣ λ_{e} ∣

then at point (

0, 0, 2

), the ratio of magnitudes of electric field values due to sheet charge to that of line charge is

π n : 1

. The value of

n

is ____ .

Answer: 16

Field of the infinite sheet:

E_{sheet} = \frac{σ _{s}}{2 ε _{0}}

(uniform).
Field of the infinite line at distance

r

E_{line} = \frac{λ _{e}}{2 π ε _{0} r}

.

The point

(0, 0, 2)

lies at

r = ∣4 - 2∣ = 2

m from the line (which sits in the

z = 4

plane parallel to

y

). With

∣ σ_{s} ∣ = 2∣ λ_{e} ∣

\frac{E _{sheet}}{E _{line}} = \frac{σ _{s} /2 ε _{0}}{λ _{e} /2 π ε _{0} r} = \frac{π σ _{s} r}{λ _{e}} = π (2) (2) = 4 π .

Given the ratio is

π n : 1

π n = 4 π \Rightarrow n = 4 \Rightarrow n = 16

.
Answer: 16.

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Integer answer