When two identical batteries of internal resistance each are connected in series across a resistor , the rate of heat produced in is . When the same batteries are connected in parallel across , the rate is . If then the value of in is

Question

When two identical batteries of internal resistance $1 Ω$ each are connected in series across a resistor $R$ , the rate of heat produced in $R$ is $J_{1}$ . When the same batteries are connected in parallel across $R$ , the rate is $J_{2}$ . If $J_{1} = 2.25 J_{2}$ then the value of $R$ in $Ω$ is [2010]

Smart Achievers · Accepted Answer

(4)

Cells connected in series

$R$

Cells connected in parallel

$J_{2} = I^{2} R = {(\frac{E}{R + \frac{r}{2}})}^{2} R$

$∴$ $J_{1} = 2.25 J_{2} (Given)$

$\frac{{(2 E)}^{2}}{{(2 r + R)}^{2}} R = 2.25 \frac{E^{2}}{{(R + \frac{r}{2})}^{2}} R$

$\Rightarrow \frac{4}{{(2 r + R)}^{2}} = \frac{2.25}{{(R + \frac{r}{2})}^{2}}$

$\Rightarrow 4 {(R + 0.5)}^{2} = 2.25 {(2 + R)}^{2} [r = 1 Ω]$

$\Rightarrow 2 (R + 0.5) = 1.5 (2 + R)$

$∴$ $R = 4 Ω$

Solution

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Solution

Q. When two identical batteries of internal resistance 1Ω each are connected in series across a resistor R, the rate of heat produced in R is J1. When the same batteries are connected in parallel across R, the rate is J2. If J1=2.25 J2 then the value of R in Ω is [2010]

Ans. (4) Cells connected in series J1=I2R=(2E2r+R)2R Cells connected in parallel J2=I2R=(ER+r2)2R ∴ J1=2.25 J2 (Given) (2E)2(2r+R)2R=2.25 E2(R+r2)2R ⇒4(2r+R)2=2.25(R+r2)2 ⇒4(R+0.5)2=2.25(2+R)2 [ r=1Ω] ⇒2(R+0.5)=1.5(2+R) ∴ R=4Ω

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