Two batteries of e.m.f 4 V and 8 V with internal resistance 1$\Omega$ and 2$\Omega$ respectively are connected in series (opposing) with a 9$\Omega$ resistor. The current and potential difference between points 'P' and 'Q' is \dots
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Always remember that while opposing batteries subtract their voltages, their internal resistances always add to the total circuit resistance. Current must still force its way physically through both batteries!
Step 1: Understanding the Question:
We have a single-loop DC circuit containing two batteries connected in opposition (positive terminal connected to positive terminal, or negative to negative) and an external resistor ($9 \, \Omega$) between points P and Q. We need to find the main circuit current and the voltage drop specifically across the external resistor. Step 2: Key Formula or Approach:
1. Net EMF: Since the batteries oppose each other, they fight to push current in opposite directions. The net EMF is the difference between them: $E_{net} = |E_1 - E_2|$.
2. Total Resistance: Series resistances always add up, regardless of battery orientation: $R_{eq} = R_{ext} + r_1 + r_2$.
3. Ohm's Law for Current: $I = \frac{E_{net}}{R_{eq}}$.
4. Potential Difference (P to Q): $V = I \times R_{ext}$. Step 3: Detailed Explanation:
1. Calculate Net EMF:
$$E_{net} = 8 \text{ V} - 4 \text{ V} = 4 \text{ V}$$
The 8V battery "wins", so it dictates the direction of the current.
2. Calculate Total Equivalent Resistance:
Internal resistances: $r_1 = 1 \, \Omega$, $r_2 = 2 \, \Omega$.
External resistance: $R = 9 \, \Omega$.
$$R_{eq} = 1 + 2 + 9 = 12 \, \Omega$$
3. Calculate the Current ($I$):
$$I = \frac{E_{net}}{R_{eq}} = \frac{4}{12} = \frac{1}{3} \text{ A}$$
4. Calculate Potential Difference between P and Q:
The only component located exactly between points P and Q is the $9 \, \Omega$ external resistor.
Using Ohm's Law across just this resistor:
$$V_{PQ} = I \times R = \left(\frac{1}{3} \text{ A}\right) \times 9 \, \Omega$$
$$V_{PQ} = 3 \text{ V}$$ Step 4: Final Answer:
The current is $1/3$ A and potential difference is 3 V, matching option (b).
