A long solenoid has a circular cross section of radius r=8.10 cm, a length ℓ=0.540 m, and n=2.

step 1 We have a long solenoid of circular cross -section of radius small r equals to 8 .10 centimeter,

A long solenoid has a circular cross section of radius...

A long solenoid has a circular cross section of radius $r=8.10 \mathrm{~cm},$ a length $\ell=0.540 \mathrm{~m}$, and $n=2.00 \cdot 10^{4}$ turns $/ \mathrm{m}$. The solenoid is carrying a current of magnitude $i=4.04 \cdot 10^{-3} \mathrm{~A}$. How much energy is stored in the magnetic field of the solenoid? a) $2.11 \cdot 10^{-7} \mathrm{~J}$ b) $8.91 \cdot 10^{-6} \mathrm{~J}$ c) $4.57 \cdot 10^{-5}$ J d) $6.66 \cdot 10^{-3}$ J. e) $4.55 \cdot 10^{-1} \mathrm{~J}$

A long solenoid has a circular cross section of radius $r=8.10 \mathrm{~cm},$ a length $\ell=0.540 \mathrm{~m}$, and $n=2.00 \cdot 10^{4}$ turns $/ \mathrm{m}$. The solenoid is carrying a current of magnitude $i=4.04 \cdot 10^{-3} \mathrm{~A}$. How much energy is stored in the magnetic field of the solenoid?
a) $2.11 \cdot 10^{-7} \mathrm{~J}$
b) $8.91 \cdot 10^{-6} \mathrm{~J}$
c) $4.57 \cdot 10^{-5}$ J
d) $6.66 \cdot 10^{-3}$ J.
e) $4.55 \cdot 10^{-1} \mathrm{~J}$

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Key Concepts

Self-Inductance and Energy Storage

Self-inductance is a property of a solenoid that quantifies its ability to store energy in its magnetic field. The inductance of a solenoid, typically expressed as L = ?? n² A ? (where A is the cross-sectional area and ? is the length), allows us to calculate the stored magnetic energy using U = 1/2 L I². This relationship links the geometry of the solenoid, the current flowing through it, and the magnetic energy it stores.

Magnetic Energy Density

Magnetic energy density defines the energy stored per unit volume in a magnetic field, and is given by u = B²/(2??). By multiplying this energy density by the volume over which the magnetic field is present, one can obtain the total energy stored in that field. This concept provides an alternative method to calculate energy storage, especially in regions with uniform magnetic fields.

Magnetic Field in a Solenoid

A long solenoid carrying an electric current produces a nearly uniform magnetic field inside it, which is given by the formula B = ?? n I, where ?? is the permeability of free space, n is the number of turns per unit length, and I is the current. This concept is fundamental to understanding how magnetic fields are created by coil geometries.

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