A metal wire of resistance R is cut into three equal pieces that are then connected side by side to form a new wire, the length of which is equal to one third of the original length. The resistance of this new wire is:-

Options

A. R
B. 3R
C. `(R)/(9)`
D. `(R)/(3)`

Answer

To solve the problem of finding the resistance of the new wire formed by cutting a metal wire of resistance \( R \) into three equal pieces and connecting them side by side, we can follow these steps:

### Step 1: Understand the Original Wire's Resistance
The resistance \( R \) of the original wire can be expressed using the formula:
\[ R = \frac{\rho L}{A} \]
where:
- \( \rho \) is the resistivity of the material,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.

### Step 2: Determine the Resistance of Each Piece
When the wire is cut into three equal pieces, each piece will have:
- Length: \( \frac{L}{3} \)
- Cross-sectional area: \( A \)

The resistance \( R_p \) of each piece can be calculated as:
\[ R_p = \frac{\rho \left(\frac{L}{3}\right)}{A} = \frac{R}{3} \]

### Step 3: Calculate the Total Resistance of the New Configuration
When the three pieces are connected side by side (in parallel), the total resistance \( R_f \) of the new wire can be calculated using the formula for resistors in parallel:
\[ \frac{1}{R_f} = \frac{1}{R_p} + \frac{1}{R_p} + \frac{1}{R_p} = 3 \cdot \frac{1}{R_p} \]
Substituting \( R_p = \frac{R}{3} \):
\[ \frac{1}{R_f} = 3 \cdot \frac{1}{\frac{R}{3}} = 3 \cdot \frac{3}{R} = \frac{9}{R} \]

### Step 4: Solve for the Total Resistance
Now, taking the reciprocal to find \( R_f \):
\[ R_f = \frac{R}{9} \]

### Conclusion
The resistance of the new wire formed by connecting the three pieces side by side is:
\[ R_f = \frac{R}{9} \]