A uniform wire of length l carrying current radiates energy at a rate P to environment while its temperature remains the same. Magnitude of magnetic flux density on the surface of wire is B and its diameter is d. If electric field strenght in wire is given by `E=(nmu_(0)P)/(Bpild)`, then find n.

Explanation

To solve the problem, we need to find the value of \( n \) in the given equation for electric field strength \( E \). Let's break down the steps systematically.

### Step 1: Understand the given parameters
We have a uniform wire of length \( l \) carrying a current \( I \) which radiates energy at a rate \( P \). The magnetic flux density on the surface of the wire is \( B \), and the diameter of the wire is \( d \).

### Step 2: Relate power, electric field, and current
The power \( P \) radiated by the wire can be expressed in terms of electric field \( E \) and current \( I \):
\[ P = E \cdot I \cdot l \]

### Step 3: Express current in terms of magnetic flux density
The magnetic field \( B \) around a long straight wire carrying current \( I \) is given by:
\[ B = \frac{\mu_0 I}{2\pi r} \]
For a wire of diameter \( d \), the radius \( r \) is \( \frac{d}{2} \). Therefore, we can express \( B \) as:
\[ B = \frac{\mu_0 I}{\pi d} \]
From this, we can solve for the current \( I \):
\[ I = \frac{B \pi d}{\mu_0} \]

### Step 4: Substitute current into the power equation
Substituting \( I \) back into the power equation:
\[ P = E \cdot \left(\frac{B \pi d}{\mu_0}\right) \cdot l \]
Rearranging gives:
\[ E = \frac{P \mu_0}{B \pi d l} \]

### Step 5: Compare with the given electric field equation
We are given that:
\[ E = \frac{n \mu_0 P}{B \pi l d} \]
Now we can set the two expressions for \( E \) equal to each other:
\[ \frac{P \mu_0}{B \pi d l} = \frac{n \mu_0 P}{B \pi l d} \]

### Step 6: Solve for \( n \)
Since \( \mu_0 \), \( P \), \( B \), \( \pi \), \( l \), and \( d \) are common in both sides of the equation, we can cancel them out:
\[ 1 = n \]
Thus, we find:
\[ n = 1 \]

### Final Answer
The value of \( n \) is \( 1 \).

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