Given:
• EMF of first cell, E1=4VE_1 = 4V, internal resistance r1=2Ωr_1 = 2 \Omega
• EMF of second cell, E2=2VE_2 = 2V, internal resistance r2=1Ωr_2 = 1 \Omega
• External resistance, R=10ΩR = 10 \Omega
Step 1: Understanding the configuration
The cells are connected in parallel, meaning the effective EMF (EeffE_{\text{eff}}) and the effective internal resistance (reffr_{\text{eff}}) need to be calculated.
Step 2: Calculate the effective EMF
For two cells connected in parallel, the effective EMF is given by:
Eeff=E1r2+E2r1r1+r2E_{\text{eff}} = \dfrac{E_1 r_2 + E_2 r_1}{r_1 + r_2}
Substituting the given values:
Eeff=(4 V)⋅(1 Ω)+(2 V)⋅(2 Ω)2 Ω+1 ΩE_{\text{eff}} = \dfrac{(4 \, \text{V}) \cdot (1 \, \Omega) + (2 \, \text{V}) \cdot (2 \, \Omega)}{2 \, \Omega + 1 \, \Omega} Eeff=4+43E_{\text{eff}} = \dfrac{4 + 4}{3} Eeff=83 V≈2.67 VE_{\text{eff}} = \dfrac{8}{3} \, \text{V} \approx 2.67 \, \text{V}
Step 3: Calculate the effective internal resistance
The effective internal resistance is given by the formula:
reff=r1r2r1+r2r_{\text{eff}} = \dfrac{r_1 r_2}{r_1 + r_2}
Substituting the values:
reff=(2 Ω)⋅(1 Ω)2 Ω+1 Ωr_{\text{eff}} = \dfrac{(2 \, \Omega) \cdot (1 \, \Omega)}{2 \, \Omega + 1 \, \Omega} reff=23 Ω≈0.67 Ωr_{\text{eff}} = \dfrac{2}{3} \, \Omega \approx 0.67 \, \Omega
Step 4: Total resistance in the circuit
The total resistance in the circuit is the sum of the effective internal resistance and the external resistance:
Rtotal=reff+RR_{\text{total}} = r_{\text{eff}} + R Rtotal=0.67 Ω+10 ΩR_{\text{total}} = 0.67 \, \Omega + 10 \, \Omega Rtotal=10.67 ΩR_{\text{total}} = 10.67 \, \Omega
Step 5: Calculate the total current in the circuit
The total current in the circuit is given by Ohm's law:
Itotal=EeffRtotalI_{\text{total}} = \dfrac{E_{\text{eff}}}{R_{\text{total}}} Itotal=2.67 V10.67 ΩI_{\text{total}} = \dfrac{2.67 \, \text{V}}{10.67 \, \Omega} Itotal≈0.25 AI_{\text{total}} \approx 0.25 \, \text{A}
Step 6: Calculate the potential difference across the external resistance
The potential difference across the external resistance RR is given by:
V=Itotal⋅RV = I_{\text{total}} \cdot R V=0.25 A⋅10 ΩV = 0.25 \, \text{A} \cdot 10 \, \Omega V=2.5 VV = 2.5 \, \text{V}
Final Answer:
The potential difference across the 10Ω10 \Omega resistor is 2.5 V2.5 \, \text{V}.
