Question

Two charged metallic spheres with radii \(R_1\) and \(R_2\) are brought in contact and then separated. The ratio of final charges \(Q_1\) and \(Q_2\) on the two spheres respectively will be _____.
Fill in the blank with the correct answer from the options given below

• A. \(\frac{Q_2}{Q_1}=\frac{R_1}{R_2}\)
• B. \(\frac{Q_2}{Q_1}<\frac{R_1}{R_2}\)
• C. \(\frac{Q_2}{Q_1}>\frac{R_1}{R_2}\)
• D. \(\frac{Q_2}{Q_1}=\frac{R_2}{R_1}\)

Answer 1

The Correct Option is D

When two charged metallic spheres with radii \(R_1\) and \(R_2\) are brought into contact and then separated, the distribution of charge will ensure that the potential on both spheres is equal after separation. The potential \(V\) of a sphere with charge \(Q\) and radius \(R\) is given by:

\(V=\frac{Q}{R}\)

After contact, let the charges be \(Q_1\) and \(Q_2\) on spheres of radii \(R_1\) and \(R_2\), respectively. Equating the potentials, we have:

\(\frac{Q_1}{R_1}=\frac{Q_2}{R_2}\)

From this equation, rearranging gives the ratio of the charges:

\(\frac{Q_2}{Q_1}=\frac{R_2}{R_1}\)

The correct answer is:

\(\frac{Q_2}{Q_1}=\frac{R_2}{R_1}\)

Answer 2

The correct answer is Option 4: Q₁/Q₂ = R₁/R₂ or Q₂/Q₁ = R₂/R₁.

Here's the explanation:

When two charged metallic spheres are brought into contact, they reach an equilibrium state where their potentials are equal. This is because charge flows from the sphere with higher potential to the sphere with lower potential until the potentials are the same.

The potential (V) of a charged sphere is given by:

V = kQ/R

Where:

• V is the potential
• k is Coulomb's constant
• Q is the charge on the sphere
• R is the radius of the sphere

When the spheres are in contact, their potentials are equal:

V₁ = V₂

Therefore:

kQ₁/R₁ = kQ₂/R₂

We can cancel k from both sides:

Q₁/R₁ = Q₂/R₂

Rearranging to find the ratio of charges:

Q₁/Q₂ = R₁/R₂

Q₂/Q₁ = R₂/R₁

Thus, the ratio of the final charges on the two spheres is directly proportional to the ratio of their radii.

Therefore, the correct answer is:

Option 4: Q₂/Q₁ = R₂/R₁