Problem

Calculate the current through $R_1$ in the circuit below:

Data

• EMF: $\varepsilon = 15.0 \, \mathrm{V}$
• Internal resistance: $r = 5.00 \, \Omega$
• Resistances: $R_1 = 100 \, \Omega$, $R_2 = 300 \, \Omega$

Prerequisite Concepts

1. Parallel Resistance:
   $\frac{1}{R'} = \frac{1}{R_1} + \frac{1}{R_2}.$
2. Ohm’s Law:
   $I = \frac{\varepsilon}{R_{\text{eq}}}.$

Solution

1. Equivalent Resistance of $R_1$ and $R_2$:
   $\frac{1}{R'} = \frac{1}{100} + \frac{1}{300} \implies R' = 75 \, \Omega.$

2. Total Resistance:
   $R_{\text{eq}} = R' + r = 75 + 5 = 80 \, \Omega.$

3. Total Current:
   $I = \frac{\varepsilon}{R_{\text{eq}}} = \frac{15}{80} = 0.1875 \, \mathrm{A}.$

4. Voltage across $R'$: $V_t = I \cdot R' = 0.1875 \cdot 75 = 14.06 \, \mathrm{V}.$

5. Current through $R_1$:
   $I_{R_1} = \frac{V_t}{R_1} = \frac{14.06}{100} = 0.1406 \, \mathrm{A}.$

Answer

• Current through $R_1$: $I_{R_1} = 0.1406 \, \mathrm{A}$

---