The magnetic energy stored in a long solenoid of area of cross-section A in a small region of length L is

Options

A. `(B^(2)AL)/(2mu_(0)^(2))`
B. `(AL)/(2mu_(0))`
C. `(1)/(2) mu_(0)B^(2)AL`
D. `(B^(2)AL)/(2mu_(0))`

Answer

To find the magnetic energy stored in a long solenoid of cross-sectional area \( A \) and length \( L \), we can follow these steps:

### Step 1: Understand the formula for energy density
The energy density \( u \) (energy per unit volume) stored in a magnetic field is given by the formula:
\[ u = \frac{B^2}{2\mu_0} \]
where \( B \) is the magnetic field strength and \( \mu_0 \) is the permeability of free space.

### Step 2: Calculate the volume of the solenoid
The volume \( V \) of the small region of the solenoid can be calculated using the formula:
\[ V = A \times L \]
where \( A \) is the cross-sectional area and \( L \) is the length of the solenoid.

### Step 3: Calculate the total magnetic energy stored
The total magnetic energy \( U \) stored in the solenoid can be calculated by multiplying the energy density by the volume:
\[ U = u \times V \]
Substituting the expressions for \( u \) and \( V \):
\[ U = \left(\frac{B^2}{2\mu_0}\right) \times (A \times L) \]

### Step 4: Simplify the expression
Now, substituting the volume into the equation:
\[ U = \frac{B^2}{2\mu_0} \times (A \times L) = \frac{B^2 A L}{2\mu_0} \]

### Final Answer
Thus, the magnetic energy stored in a long solenoid of cross-sectional area \( A \) in a small region of length \( L \) is given by:
\[ U = \frac{B^2 A L}{2\mu_0} \]