Consider a copper cylinder of volume x, resistivity f, resistance across its length is r. If diameter of cylindrical conductor is `((16fx)/(pi^(2)r))^(1//k)` , then find the value of k.

Answer

To solve the problem, we need to find the value of \( k \) given the diameter of a copper cylinder in terms of its volume, resistivity, and resistance. Let's break down the solution step by step.

### Step 1: Understand the given parameters
We have:
- Volume of the cylinder: \( V = x \)
- Resistivity of the material: \( \rho = f \)
- Resistance across its length: \( R = r \)
- Diameter of the cylinder: \( d = \left( \frac{16fx}{\pi^2 r} \right)^{\frac{1}{k}} \)

### Step 2: Relate volume to diameter and length
The volume \( V \) of a cylinder can be expressed as:
\[ V = A \cdot l \]
where \( A \) is the cross-sectional area and \( l \) is the length of the cylinder. The cross-sectional area \( A \) for a cylinder with diameter \( d \) is given by:
\[ A = \frac{\pi d^2}{4} \]
Thus, we can write:
\[ x = \frac{\pi d^2}{4} \cdot l \]

### Step 3: Solve for length \( l \)
Rearranging the equation for volume, we get:
\[ l = \frac{4x}{\pi d^2} \]

### Step 4: Use the formula for resistance
The resistance \( R \) of a cylindrical conductor is given by:
\[ R = \frac{\rho l}{A} \]
Substituting the expressions for \( l \) and \( A \):
\[ R = \frac{f \cdot \left( \frac{4x}{\pi d^2} \right)}{\frac{\pi d^2}{4}} \]
This simplifies to:
\[ R = \frac{4fx}{\pi d^2} \cdot \frac{4}{\pi d^2} = \frac{16fx}{\pi^2 d^2} \]

### Step 5: Rearranging to find \( d^2 \)
From the equation \( R = \frac{16fx}{\pi^2 d^2} \), we can express \( d^2 \) in terms of \( R \):
\[ d^2 = \frac{16fx}{\pi^2 R} \]

### Step 6: Find \( d \)
Taking the square root of both sides gives:
\[ d = \sqrt{\frac{16fx}{\pi^2 R}} = \left( \frac{16fx}{\pi^2 R} \right)^{\frac{1}{2}} \]

### Step 7: Compare with the given diameter expression
We are given:
\[ d = \left( \frac{16fx}{\pi^2 r} \right)^{\frac{1}{k}} \]
From our derivation, we have:
\[ d = \left( \frac{16fx}{\pi^2 R} \right)^{\frac{1}{2}} \]

### Step 8: Equate the two expressions for \( d \)
Setting the two expressions for \( d \) equal gives:
\[ \left( \frac{16fx}{\pi^2 R} \right)^{\frac{1}{2}} = \left( \frac{16fx}{\pi^2 r} \right)^{\frac{1}{k}} \]

### Step 9: Solve for \( k \)
From the equality, we can deduce:
\[ \frac{1}{2} = \frac{1}{k} \]
Thus, solving for \( k \) gives:
\[ k = 4 \]

### Final Answer
The value of \( k \) is \( 4 \).

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