Three conductors draw respectively currents of 1 A, 2 A and 4 A when connected in turn across a battery. If they are connected in series across the same battery, the current drawn will be

Options

A. `(2)/(7)A`
B. `(3)/(7)A`
C. `(4)/(7)A`
D. `(5)/(7)A`

Answer

To solve the problem step by step, we need to find the equivalent resistance of the three conductors when connected in series and then use Ohm's law to find the current drawn from the battery.

### Step 1: Determine the Resistance of Each Conductor
When a conductor is connected across a battery, the resistance can be calculated using Ohm's law, which states that \( R = \frac{V}{I} \).

1. For the first conductor (current = 1 A):
\[ R_1 = \frac{V}{1} = V \]

2. For the second conductor (current = 2 A):
\[ R_2 = \frac{V}{2} \]

3. For the third conductor (current = 4 A):
\[ R_3 = \frac{V}{4} \]

### Step 2: Calculate the Equivalent Resistance in Series
When resistors are connected in series, the total or equivalent resistance \( R_{eq} \) is the sum of the individual resistances:
\[ R_{eq} = R_1 + R_2 + R_3 \]
Substituting the values we found:
\[ R_{eq} = V + \frac{V}{2} + \frac{V}{4} \]

### Step 3: Find a Common Denominator
To add these fractions, we need a common denominator. The least common multiple of 1, 2, and 4 is 4. Thus, we can rewrite the resistances:
\[ R_{eq} = \frac{4V}{4} + \frac{2V}{4} + \frac{V}{4} = \frac{4V + 2V + V}{4} = \frac{7V}{4} \]

### Step 4: Calculate the Current Drawn from the Battery
Using Ohm's law again, the current \( I \) drawn from the battery can be calculated as:
\[ I = \frac{V}{R_{eq}} \]
Substituting the equivalent resistance:
\[ I = \frac{V}{\frac{7V}{4}} = V \cdot \frac{4}{7V} = \frac{4}{7} \text{ A} \]

### Final Answer
The current drawn when the three conductors are connected in series across the same battery is:
\[ \frac{4}{7} \text{ A} \]

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