Let , and be the respective electric fields at a distance from a point charge , an infinitely long wire with constant linear charge density , and an infinite plane with uniform surface charge density . If at a given distance , then

Question

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 [2014]

Smart Achievers · Accepted Answer

(3)

$E_{1} (r_{0}) = E_{2} (r_{0}) = E_{3} (r_{0}) (Given)$

$(1) E_{1} (r_{0}) = E_{3} (r_{0})$ $∴$ $\frac{1}{4 π ε_{0}} \frac{Q}{r_{0}^{2}} = \frac{σ}{2 ε_{0}}$ $\Rightarrow Q = 2 π σ r_{0}^{2}$

$(2) E_{2} (r_{0}) = E_{3} (r_{0})$ $\frac{1}{4 π ε_{0}} \frac{2 λ}{r_{0}} = \frac{σ}{2 ε_{0}}$ $\Rightarrow r_{0} = \frac{λ}{σ π}$

$(3) E_{1} (\frac{r_{0}}{2}) = \frac{1}{4 π ε_{0}} \frac{4 Q}{r_{0}^{2}}$

$= \frac{1}{4 π ε_{0}} \times \frac{4 \times 2 λ r_{0}}{r_{0}^{2}} = \frac{1}{4 π ε_{0}} \frac{8 λ}{r_{0}}$

$∵$ $E_{1} (r_{0}) = E_{2} (r_{0})$ $\Rightarrow Q = 2 λ r_{0}$

$And$ $2 E_{2} (\frac{r_{0}}{2}) = 2 [\frac{1}{4 π ε_{0}} \frac{4 λ}{r_{0}}] = \frac{1}{4 π ε_{0}} \frac{8 λ}{r_{0}}$

$(4) E_{2} (\frac{r_{0}}{2}) = \frac{1}{4 π ε_{0}} \frac{2 λ}{r_{0} / 2} = \frac{1}{4 π ε_{0}} \frac{4 λ}{r_{0}} = \frac{λ}{π ε_{0} r_{0}}$

$4 E_{3} (\frac{r_{0}}{2}) = 4 \frac{σ}{2 ε_{0}} = \frac{2 σ}{ε_{0}} = \frac{2}{ε_{0}} \cdot \frac{λ}{π r_{0}}$

$= \frac{2 λ}{π ε_{0} r_{0}}$

Solution

1 $Q = 4 σ π r_{0}^{2}$

2 $r_{0} = \frac{λ}{2 π σ}$

3 $E_{1} (\frac{r_{0}}{2}) = 2 E_{2} (\frac{r_{0}}{2})$

4 $E_{2} (\frac{r_{0}}{2}) = 4 E_{3} (\frac{r_{0}}{2})$

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Solution

Q. Let E1(r), E2(r) and E3(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 E1(r0)=E2(r0)=E3(r0) at a given distance r0, then [2014]

1 Q=4σπr02

2 r0=λ2πσ

3 E1(r02)=2 E2(r02)

4 E2(r02)=4 E3(r02)

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1 $Q = 4 σ π r_{0}^{2}$

2 $r_{0} = \frac{λ}{2 π σ}$

3 $E_{1} (\frac{r_{0}}{2}) = 2 E_{2} (\frac{r_{0}}{2})$

4 $E_{2} (\frac{r_{0}}{2}) = 4 E_{3} (\frac{r_{0}}{2})$