JEE Advanced · Physics · 17. Electrostatics

Two non-conducting spheres of radii $R_{1}$ and $R_{2}$ and carrying uniform volume charge densities $+ ρ$ and $- ρ$ , respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region,

A the electrostatic field is zero.
B the electrostatic potential is constant.
C the electrostatic field is constant in magnitude.
D the electrostatic field has same direction.

Verified Solution

Answer & Solution

Correct Answer

(D) the electrostatic field has same direction.

Step-by-step Solution

Detailed explanation

In triangle

P C_{1} C_{2}

{\vec{r}}_{2} = \vec{d} + {\vec{r}}_{1}

The electrostatic field at point

P

is,

\vec{E} = \frac{K (ρ \frac{4}{3} π R_{1}^{3}) ({\vec{r}}_{2})}{R_{1}^{3}} + \frac{K (ρ \frac{4}{3} π R_{2}^{3}) (- {\vec{r}}_{1})}{R_{2}^{3}}

\vec{E} = K ρ \frac{4}{3} π ({\vec{r}}_{2} - {\vec{r}}_{1})

\vec{E} = \frac{ρ}{3 ε_{0}} \vec{d}

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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Columns 1, 2 and 3 contain starting materials, reaction conditions and type of reactions respectively.

Column 1	Column 2	Column 3
(I) Toulene	(i) $NaOH$ $ /Br _2$	(P) Condensation
(II) Acetophenone	(ii) $B r_2 / h v$	(Q) Carboxylation
(III) Benzaldehyde	(iii) $\left( CH _3 CO \right)_2$ $O / CH _3$ $COOK$	(R) Substitution
(IV) Phenol	(iv) $NaOH$ $/CO _2$	(S) Haloform

The only CORRECT combination in which the reaction proceeds through radical mechanism is

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Two non-conducting spheres of radii R1 and R2 and carrying uniform volume charge densities +ρ and -ρ, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region,

Step-by-step Solution

Two non-conducting spheres of radii $R_{1}$ and $R_{2}$ and carrying uniform volume charge densities $+ ρ$ and $- ρ$ , respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region,