Two concentric coplanar circular loops of wire (resistance per unit length 10 -4 ohms m-1) have diameters 0.2 m and 2 m. With time-varying potential difference of (4+2.5t) applied to larger loop, find current in smaller one.

Magnetic field at O (centre) due to larger loop’s current equals = B = μ₀I/2R

Now, assuming that r is resistance per unit length, then the following can be said:

I = potential difference/resistance = 4 + 2.5t/2πR⋅ρ \(\times\) πr²

\(\therefore\) Thus, it can also be expressed as:

⇒ B = μ₀/2R \(\times\) 4 + 2.5t/2πR . ρ

⇒ r < ϕ = B × πr² = μ₀/2R \(\times\) 4 + 2.5t/2πR . ρ \(\times\) πr²

Thus, the Induced emf can be expressed as, e = dϕ/dt

= μ₀r²/4R²ρ \(\times\) 2.5

Corresponding current in smaller loop is = I′ 

Therefore,

⇒ I′ = e/R = μ₀r²/4R²ρ × 2.5 × 1/2πrρ

= 2.5μ₀r/8πR²ρ² 

= 2.5 \(\times\) 4π \(\times\) 10⁻⁷ \(\times\) 0.1/8π \(\times\) (1)² \(\times\) (10⁻⁴)²

\(\therefore\) I′ = 1.25 A

Hence, the current in smaller loop is 1.25 A

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