Surface Charge Density After Contact – Ratio σ₁/σ₂ | JEE Electrostatics | Doubtify JEE

 

⚡ Surface Charge Density After Contact – Electrostatics | JEE Physics | Doubtify JEE

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📌 Question:

Two metal spheres of radius R and 3R have the same surface charge density σ.
They are brought in contact and then separated.
After separation, let the surface charge densities be σ₁ (for smaller sphere) and σ₂ (for larger sphere).
What is the ratio σ₁ / σ₂?

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🖼️ Question Image:

🧠 Solution Image:

🧪 Detailed Concept & Solution:

⚙️ Given:

• Radii: R (small sphere) and 3R (large sphere)

• Initial surface charge density: σ on both spheres
  ⇒ So initial charges:
  $Q_1 = \sigma \cdot 4\pi R^2$
  $Q_2 = \sigma \cdot 4\pi (3R)^2 = 9\sigma \cdot 4\pi R^2$
  

🔁 When they are brought in contact:

• Charge redistributes such that potential becomes equal
  For a sphere,
  $V = \dfrac{1}{4\pi \varepsilon_0} \cdot \dfrac{Q}{R}$

Let final charges be Q₁′ and Q₂′ after contact.

So:

$$\frac{Q_1'}{R} = \frac{Q_2'}{3R} \Rightarrow Q_2' = 3Q_1'$$

Also, total charge is conserved:

$$Q_1' + Q_2' = Q_1 + Q_2 = \sigma \cdot 4\pi R^2 (1 + 9) = 10\sigma \cdot 4\pi R^2$$

Substitute Q₂′ = 3Q₁′:

$$Q_1' + 3Q_1' = 4Q_1' = 10\sigma \cdot 4\pi R^2 \Rightarrow Q_1' = 2.5\sigma \cdot 4\pi R^2$$

So:

$$Q_2' = 7.5\sigma \cdot 4\pi R^2$$

✅ Now, Surface Charge Density:

$$σ_1 = \frac{Q_1'}{4\pi R^2} = 2.5σ \quad σ_2 = \frac{Q_2'}{4\pi (3R)^2} = \frac{7.5σ \cdot 4\pi R^2}{4\pi 9R^2} = \frac{7.5σ}{9}$$

So:

$$\frac{σ_1}{σ_2} = \frac{2.5σ}{\frac{7.5σ}{9}} = \frac{2.5 \cdot 9}{7.5} = \frac{22.5}{7.5} = 3$$

✅ Final Answer:

$$\boxed{σ_1 / σ_2 = 3}$$

🎥 Video Solution:

🔍 Why This Question is Important:

• Conceptual mix of Electrostatics + Charge Sharing

• Tests your understanding of potential equalization and surface area-based reasoning

• Commonly asked in JEE Main and Advanced

• Mastering this helps in solving problems involving charge redistribution in conductors

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🧠 Pro Tip:

Whenever two conducting spheres are connected, their potentials equalize — not charge or surface density. Always start from that principle.

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