Answer :
Answer:
The magnitude of the magnetic field at the center of a solenoid is 80 times of the magnetic field at the center of a loop.
Explanation:
Given that,
Radius = 20 mm
Number of turns = 2000
Suppose, we can use wire of 1.0 mm diameter to make either a single loop of wire or a solenoid, and we wish to compare, for a given current, the magnetic fields at the center of each.
Suppose a small element of wire of length dl.
Let I be the current flowing through the wire
We need to calculate the magnetic field
Using formula of magnetic field
[tex]B=\int{\dfrac{\mu_{0}}{4\pi}\times\dfrac{I\times dl}{R^2}}[/tex]
[tex]B=\dfrac{\mu_{0}I}{4\pi\times R^2}\int{dl}[/tex]
Put the value into the formula
[tex]B=\dfrac{\mu_{0}I}{4\pi\times r^2}\times2\pi R[/tex]
[tex]B=\dfrac{\mu_{0}I}{2R}[/tex]...(I)
We need to calculate the magnetic field due to solenoid
Using formula of magnetic field
[tex]B'=\mu_{0}nI[/tex]...(II)
We need to calculate the magnitude of the magnetic field at the center of a circular current loop
Dividing equation (II) by equation (I)
[tex]\dfrac{B'}{B}=\dfrac{\mu_{0}nI}{\dfrac{\mu_{0}I}{2R}}[/tex]
[tex]\dfrac{B'}{B}=2nR[/tex]
Put the value into the formula
[tex]\dfrac{B'}{B}=2\times2000\times20\times10^{-3}[/tex]
[tex]\dfrac{B'}{B}=80[/tex]
[tex]B'=80 B[/tex]
Hence, The magnitude of the magnetic field at the center of a solenoid is 80 times of the magnetic field at the center of a loop.
The magnitude of the magnetic field at the center of a solenoid is [tex]80[/tex]times of the magnetic field at the center of a loop.
Magnetic field strength :
The magnitude of the magnetic field at the center of a circular current loop is given as,
[tex]B_{L}=\frac{\mu _{0}I}{2R}[/tex]
the magnetic field at the center of a solenoid is,
[tex]B_{S}=\mu_{0}nI[/tex]
Where
- [tex]\mu_{0}[/tex] magnetic permeability
- [tex]I[/tex] is current
- [tex]R[/tex] is radius.
Taking the ratio of both
[tex]\frac{B_{S}}{B_{L}}=2nR[/tex]
Given that, [tex]n=2000,R=20mm=20*10^{-3}m[/tex]
[tex]\frac{B_{S}}{B_{L}}=2*2000*20*10^{-3} \\\\\frac{B_{S}}{B_{L}}=80\\\\B_{s}=80B_{L}[/tex]
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