The magnetic field along the axis of an air cored solenoid is B. The magnetic field energy density is

Options

A. `1/2 (B^2)/(mu_0)`
B. `1/2 mu_0 B^2`
C. `1/2 mu_0 B`
D. `(B)/(2mu_0)`

Answer

To find the magnetic field energy density in an air-cored solenoid, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Magnetic Field Energy Density**:
The magnetic field energy density (u_B) is defined as the energy (E) stored in a magnetic field per unit volume (V). The formula is given by:
\[ u_B = \frac{E}{V} \]

2. **Expression for Energy in a Solenoid**:
The energy stored in the magnetic field of a solenoid can be expressed as:
\[ E = \frac{1}{2} L I^2 \]
where \(L\) is the inductance of the solenoid and \(I\) is the current flowing through it.

3. **Volume of the Solenoid**:
The volume (V) of the solenoid can be calculated as:
\[ V = A \cdot l \]
where \(A\) is the cross-sectional area and \(l\) is the length of the solenoid.

4. **Inductance of the Solenoid**:
The inductance \(L\) of an air-cored solenoid is given by:
\[ L = \mu \frac{n^2 A}{l} \]
where \(\mu\) is the permeability of free space, \(n\) is the number of turns per unit length, and \(A\) is the cross-sectional area.

5. **Magnetic Field Strength**:
The magnetic field strength \(B\) in the solenoid is given by:
\[ B = \mu n I \]
From this, we can express the current \(I\) as:
\[ I = \frac{B l}{\mu n} \]

6. **Substituting Values into the Energy Equation**:
Now, substituting the expression for \(I\) into the energy equation:
\[ E = \frac{1}{2} L I^2 = \frac{1}{2} \left(\mu \frac{n^2 A}{l}\right) \left(\frac{B l}{\mu n}\right)^2 \]

7. **Simplifying the Energy Expression**:
Simplifying the above expression:
\[ E = \frac{1}{2} \left(\mu \frac{n^2 A}{l}\right) \left(\frac{B^2 l^2}{\mu^2 n^2}\right) = \frac{1}{2} \frac{B^2 A l}{\mu} \]

8. **Finding the Energy Density**:
Now, substituting \(E\) and \(V\) into the energy density formula:
\[ u_B = \frac{E}{V} = \frac{\frac{1}{2} \frac{B^2 A l}{\mu}}{A l} = \frac{1}{2} \frac{B^2}{\mu} \]

9. **Final Expression for Magnetic Field Energy Density**:
Thus, the magnetic field energy density is:
\[ u_B = \frac{B^2}{2\mu} \]

### Conclusion:
The magnetic field energy density in an air-cored solenoid is given by:
\[ u_B = \frac{B^2}{2\mu} \]