JEE Advanced · Physics · 17. Electrostatics

Let $E_{1} (r), E_{2} (r)$ and $E_{3} (r)$ be the respective electric fields at a distance r from a point charge Q, an infinitely long wire with constant linear charge density $λ$ , and an infinite plane with uniform surface charge density $σ$ . If $E_{1} (r_{0}) = E_{2} (r_{0}) = E_{3} (r_{0})$ at a given distance $r_{0}$ , then

A $Q = 4 σ π r_{0}^{2}$
B $r_{0} = \frac{λ}{2 π σ}$
C $E_{1} (\frac{r_{0}}{2}) = 2 E_{2} (\frac{r_{0}}{2})$
D $E_{2} (\frac{r_{0}}{2}) = 4 E_{3} (\frac{r_{0}}{2})$

Verified Solution

Answer & Solution

Correct Answer

(C) $E_{1} (\frac{r_{0}}{2}) = 2 E_{2} (\frac{r_{0}}{2})$

Step-by-step Solution

Detailed explanation

\frac{Q}{4 π ε_{0} r_{0}^{2}} = \frac{λ}{2 π ε_{0} r_{0}} = \frac{σ}{2 ε_{0}}

E_{1} (\frac{r_{0}}{2}) = \frac{Q}{π ε_{0} r_{0}^{2}}, E_{2} (\frac{r_{0}}{2}) = \frac{λ}{π ε_{0} r_{0}}, E_{3} (\frac{r_{0}}{2}) = \frac{σ}{2 ε_{0}}

E_{1} (\frac{r_{0}}{2}) = 2 E_{2} (\frac{r_{0}}{2})

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

Column I	Column II
(A) The minimum value of $\frac{x^2+2 x+4}{x+2}$ is	(p) 0
(B) Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skew-symmetric and $(A+B)(A-B) =$ $(A-B)(A+B)$. If $(A B)^t=(-1)^k$ $A B$, where $(A B)^t$ is the transpose of the matrix $A B$, then possible values of $k$ are	(q) 1
(C) Let $a=\log _3 \log _3 2$. An integer $k$ satisfying $1<2^{\left(-k+3^{-a}\right)}<2$, must be less than	(r) 2
(D) If $\sin \theta=\cos \phi$, then the possible values of $\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)$ are	(s) 3

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Column I	Column II
(A) The minimum value of \(\frac{x^2+2 x+4}{x+2}\) is	(p) 0
(B) Let \(A\) and \(B\) be \(3 \times 3\) matrices of real numbers, where \(A\) is symmetric, \(B\) is skew-symmetric and \((A+B)(A-B) =\) \((A-B)(A+B)\). If \((A B)^t=(-1)^k\) \(A B\), where \((A B)^t\) is the transpose of the matrix \(A B\), then possible values of \(k\) are	(q) 1
(C) Let \(a=\log _3 \log _3 2\). An integer \(k\) satisfying \(1<2^{\left(-k+3^{-a}\right)}<2\), must be less than	(r) 2
(D) If \(\sin \theta=\cos \phi\), then the possible values of \(\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)\) are	(s) 3

Let E1r, E2r and E3r be the respective electric fields at a distance r from a point charge Q, an infinitely long wire with constant linear charge density λ , and an infinite plane with uniform surface charge density σ . If E1r0=E2r0=E3r0 at a given distance r0 , then

Step-by-step Solution