Two cells of 1.5 V and 2 V , having internal resistances of `1Omega` and `2Omega` respectively, are connected in parallel so as to read the current in the same direction through an external resistance of `5 Omega` . The current in the external resistance will be

Answer

To solve the problem of finding the current through the external resistance when two cells are connected in parallel, we can follow these steps:

### Step 1: Identify the components and their values
We have two cells:
- Cell 1: EMF (E1) = 1.5 V, Internal Resistance (R1) = 1 Ω
- Cell 2: EMF (E2) = 2 V, Internal Resistance (R2) = 2 Ω
- External Resistance (R3) = 5 Ω

### Step 2: Calculate the equivalent EMF (V_AB) and equivalent internal resistance (R_eq)
When cells are connected in parallel, the equivalent EMF (V_AB) can be calculated using the formula:

\[ \frac{V_{AB}}{R_{eq}} = \frac{E_1}{R_1} + \frac{E_2}{R_2} \]

Where \( R_{eq} \) is the equivalent internal resistance given by:

\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]

### Step 3: Calculate the equivalent internal resistance (R_eq)
Using the values:

\[ \frac{1}{R_{eq}} = \frac{1}{1} + \frac{1}{2} = 1 + 0.5 = 1.5 \]

Thus,

\[ R_{eq} = \frac{1}{1.5} = \frac{2}{3} \, \Omega \]

### Step 4: Calculate the equivalent EMF (V_AB)
Now, substituting the values into the EMF equation:

\[ \frac{V_{AB}}{R_{eq}} = \frac{1.5}{1} + \frac{2}{2} \]

Calculating the right side:

\[ \frac{V_{AB}}{R_{eq}} = 1.5 + 1 = 2.5 \]

Now substituting \( R_{eq} \):

\[ V_{AB} = 2.5 \times R_{eq} = 2.5 \times \frac{2}{3} = \frac{5}{3} \, V \]

### Step 5: Calculate the total current (I) through the external resistance (R3)
Using Ohm's law, the current through the external resistance can be calculated as:

\[ I = \frac{V_{AB}}{R3} \]

Substituting the values:

\[ I = \frac{\frac{5}{3}}{5} = \frac{5}{15} = \frac{1}{3} \, A \]

### Final Answer
The current in the external resistance will be \( \frac{1}{3} \, A \).

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