A cell is connected across two points P and Q of a uniform circular conductor. Prove that the magnetic field at its centre O will be zero. 

Let I₁, I₂ be the lengths of the two parts PRQ and PSQ of the circular conductor and ρ be the resistance per unit length of this conductor. Let r be the radius of circular conductor. 

Resistance of the portion PRQ will be R₁ = ρl₁ = ρrθ₁ 

Resistance of the portion PSQ will be R₂ = ρl₂ = ρrθ₂      

Let l₁ and l₂ be the currents in the portion PRQ and PSQ respectively. The potential difference across P and Q is 

V_P – V_Q = l₁R₁ = l₂R₂

or,   l₁ρrθ₁ = l₂ρrθ₂

or, l₁θ₁ = l₂θ₂                             … (i)