Chapter 4: Q19E (page 1014)

: A solenoid $25 c m$ long and with a cross-sectional area of $0.5 c m$ contains $400$ turns of wire and carries a current of $80 A$ . Calculate: (a) the magnetic field in the solenoid; (b) the energy density in the magnetic field if the solenoid is filled with air; (c) the total energy contained in the coil’s magnetic field (assume the field is uniform); (d) the inductance of the solenoid.

Short Answer

Expert verified

a) $B = 0.161 T$

b) $u = 1.03 \times 10^{4} J / m^{3}$

c) $U = 129 m J$

d) $L = 4.02 \times 10^{- 5} H$

Step by step solution

The formulas used in the given problem

The magnetic field in a solenoid is given by $B = μ_{0} n I$ .

The energy stored in a solenoid is given by $U = \frac{1}{2 L I^{2}}$ .

The energy density is given by $\frac{B^{2}}{2 μ_{0}}$ or it is simple energy per volume.

Calculate the magnetic field

The magnetic field is given by

$B = μ_{0} n I B = \frac{μ_{0} N I}{l} B = \frac{4 π \times 10^{- 7} \times 400 \times 80}{0.25} B = 0.161 T$

Calculate the energy density

The energy density is given by

$u = \frac{B^{2}}{2 μ_{0}} u = \frac{0 . 161^{2}}{2 \times 4 π \times 10^{- 7}} u = 1.03 \times 10^{4} J / m^{3}$

Calculate the total energy stored in the solenoid

The total energy can be given by the product of energy density and volume.

$V = l A V = 0.25 \times 0.5 \times 10^{- 4} V = 1.25 \times 10^{- 5} m^{3}$

The total energy will be

$U = u V U = 1.03 \times 10^{4} \times 1.25 \times 10^{- 5} U = 129 m J$

Calculate the self-inductance

The energy is given by $U = \frac{1}{2} L I^{2}$ . Use this formula to calculate the self-inductance.

$L = \frac{2 U}{I^{2}} L = \frac{2 \times 0.129}{80^{2}} L = 4.02 \times 10^{- 5} H$

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Short Answer

Step by step solution

The formulas used in the given problem

Calculate the magnetic field

Calculate the energy density

Calculate the total energy stored in the solenoid

Calculate the self-inductance

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