The potential difference across a cell is 1.8 V when a current of 0.5 A is drawn from it. The p.d. falls to 1.6 V when a current of 1.0 A is drawn. Find the emf and the internal resistance of the cell.

Answer

To find the electromotive force (emf) and the internal resistance of the cell, we can use the following steps:

### Step 1: Understand the given information
We have two scenarios:
1. When a current \( I_1 = 0.5 \, \text{A} \) is drawn, the potential difference \( V_1 = 1.8 \, \text{V} \).
2. When a current \( I_2 = 1.0 \, \text{A} \) is drawn, the potential difference \( V_2 = 1.6 \, \text{V} \).

### Step 2: Write the equations using Ohm's Law
According to Ohm's law, the relationship between emf (E), potential difference (V), current (I), and internal resistance (r) is given by:
\[ V = E - I \cdot r \]

For the first scenario:
\[ V_1 = E - I_1 \cdot r \]
Substituting the values:
\[ 1.8 = E - 0.5r \quad \text{(Equation 1)} \]

For the second scenario:
\[ V_2 = E - I_2 \cdot r \]
Substituting the values:
\[ 1.6 = E - 1.0r \quad \text{(Equation 2)} \]

### Step 3: Solve the equations
Now we have two equations:
1. \( 1.8 = E - 0.5r \)
2. \( 1.6 = E - 1.0r \)

We can rearrange both equations to express E in terms of r:
From Equation 1:
\[ E = 1.8 + 0.5r \quad \text{(Equation 3)} \]

From Equation 2:
\[ E = 1.6 + 1.0r \quad \text{(Equation 4)} \]

### Step 4: Set the equations for E equal to each other
Since both equations equal E, we can set them equal to each other:
\[ 1.8 + 0.5r = 1.6 + 1.0r \]

### Step 5: Solve for r
Rearranging the equation gives:
\[ 1.8 - 1.6 = 1.0r - 0.5r \]
\[ 0.2 = 0.5r \]
\[ r = \frac{0.2}{0.5} = 0.4 \, \Omega \]

### Step 6: Substitute r back to find E
Now we can substitute \( r \) back into either Equation 3 or Equation 4 to find E. Using Equation 3:
\[ E = 1.8 + 0.5 \times 0.4 \]
\[ E = 1.8 + 0.2 = 2.0 \, \text{V} \]

### Final Results
- The emf \( E \) of the cell is \( 2.0 \, \text{V} \).
- The internal resistance \( r \) of the cell is \( 0.4 \, \Omega \).