Consider a uniform spherical charge distribution of radius $R_{1}$ centred at the origin $O$ . In this distribution, a spherical cavity of radius $R_{2}$ , centred at $P$ with distance $O P = a = R_{1} - R_{2}$ (see figure), is made. If the electric field inside the cavity at position $\vec{r}$ is $\vec{E} (\vec{r})$ , then the correct statement(s) is (are) [2015]

Question

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(4)

Let us consider a point M inside the cavity where the electric field has to be calculated. Assume the cavity contains a similar charge distribution of positive and negative charge as the rest of the sphere. Electric field at M due to uniformly distributed charge of the whole sphere of radius $R_{1}$

${\vec{E}}_{1} = \frac{ρ}{3 ε} \vec{r}$

Electric field at M due to the negative charge distribution in the cavity

${\vec{E}}_{2} = \frac{ρ}{3 ε} \vec{M P}$

$∴$ The total electric field at M inside the cavity

$\vec{E} = {\vec{E}}_{1} + {\vec{E}}_{2} = \frac{ρ}{3 ε} \vec{r} + \frac{ρ}{3 ε} \vec{M P}$

$∴$ $\vec{E} = \frac{ρ}{3 ε} \vec{r} + \frac{ρ}{3 ε} (\vec{a} - \vec{r})$ $[∵ \vec{r} + \vec{M P} = \vec{a}]$

$∴$ $\vec{E} = \frac{ρ}{3 ε} \vec{a}$

Hence inside the cavity $\vec{E}$ is uniform and both its magnitude and direction depend on $\vec{a}$ .

Solution

1 $\vec{E}$ is uniform, its magnitude is independent of $R_{2}$ , but its direction depends on $\vec{r}$ .

2 $\vec{E}$ is uniform, its magnitude depends on $R_{2}$ , and its direction depends on $\vec{r}$ .

3 $\vec{E}$ is uniform, its magnitude is independent of $a$ , but its direction depends on $\vec{a}$ .

4 $\vec{E}$ is uniform and both its magnitude and direction depend on $\vec{a}$ .

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Solution

1 E→ is uniform, its magnitude is independent of R2, but its direction depends on r→.

2 E→ is uniform, its magnitude depends on R2, and its direction depends on r→.

3 E→ is uniform, its magnitude is independent of a, but its direction depends on a→.

4 E→ is uniform and both its magnitude and direction depend on a→.

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1 $\vec{E}$ is uniform, its magnitude is independent of $R_{2}$ , but its direction depends on $\vec{r}$ .

2 $\vec{E}$ is uniform, its magnitude depends on $R_{2}$ , and its direction depends on $\vec{r}$ .

3 $\vec{E}$ is uniform, its magnitude is independent of $a$ , but its direction depends on $\vec{a}$ .

4 $\vec{E}$ is uniform and both its magnitude and direction depend on $\vec{a}$ .