Question

Under what condition current passing through the resistance R can be increased by short circuiting the battery of emf E₂. The internal resistances of the two batteries are r₁ and r₂ respectively.

• A. E₂r₁>E₁(R+r₂)
• B. E₁r₂>E₂(R+r₁)
• C. E₂r₂>E₁(R+r₂)
• D. E₁r₁>E₂(R+r₁)

Answer 1

Answer: (B)

E₁r₂>E₂(R+r₁)

Answer 2

Assuming batteries are in series aiding, initial current $I_1 = \frac{E_1 + E_2}{R + r_1 + r_2}$. Short circuiting battery E₂ means replacing the branch E₂-r₂ with a short circuit. Then the circuit consists of E₁ and r₁ in series with R. The current is $I_2 = \frac{E_1}{R + r_1}$. We want $I_2 > I_1$.

$\frac{E_1}{R + r_1} > \frac{E_1 + E_2}{R + r_1 + r_2}$

$E_1 (R + r_1 + r_2) > (E_1 + E_2) (R + r_1)$

$E_1 R + E_1 r_1 + E_1 r_2 > E_1 R + E_1 r_1 + E_2 R + E_2 r_1$

$E_1 r_2 > E_2 R + E_2 r_1 = E_2 (R + r_1)$.

Therefore, the condition is $E_1 r_2 > E_2 (R + r_1)$.