The electric field at r = R is [2008]

Question

The nuclear charge (Ze) is non-uniformly distributed within a nucleus of radius R. The charge density $ρ (r)$ [charge per unit volume] is dependent only on the radial distance $r$ from the centre of the nucleus as shown in figure. The electric field is only along the radial direction. [2008]

Q. The electric field at r = R is

Smart Achievers · Accepted Answer

(1)

At $r = R$ , area $A = 4 π R^{2}$ and total enclosed charge, $q_{in} = z e$

From Gauss's theorem $ϕ = \frac{q_{in}}{ε_{0}}$

or, $E A = \frac{q_{in}}{ε_{0}}$ or, $E (4 π R^{2}) = \frac{z e}{ε_{0}}$

$∴$ $E = \frac{1}{4 π ε_{0}} \frac{z e}{R^{2}}$

$Clearly electric field E is independent of A .$

Solution

1 independent of $a$

2 directly proportional to $a$

3 directly proportional to $a^{2}$

4 inversely proportional to $a$

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Solution

1 independent of a

2 directly proportional to a

3 directly proportional to a2

4 inversely proportional to a

Ans. (1) At r=R, area A=4πR2 and total enclosed charge, qin=ze From Gauss's theorem ϕ=qinε0 or, EA=qinε0 or, E(4πR2)=zeε0 ∴ E=14πε0zeR2 Clearly electric field E is independent of A.

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1 independent of $a$

2 directly proportional to $a$

3 directly proportional to $a^{2}$

4 inversely proportional to $a$