Dear sohan,
The surface charge density = σ = charge per m^2
Charge = σ * area of the inner surface
The area of the inner surface = 2 * π * r * l
l = length of cylinder
Charge = σ * 2 * π * r * l Coulombs
If we let the length of the cylinder = 1 meter,
The amount of charge on a 1 meter length of the cylinder = σ * 2 * π * r
1 amp of current = 1 coulomb per second
When the angular velocity = ω, the cylinder is rotating ω radians per second.
1 radian per second means a point on the inner surface is moving at a linear velocity of 1 radius per second. So the charged particles on the inner surface is moving at a linear velocity of (ω * r) meters per second.
This means the charge on a 1 meter length of the cylinder is moving at a velocity of (ω * r) meters per second.
The charge on a 1 meter length of the cylinder = σ * 2 * π * r
(σ * 2 * π * r) Coulombs of charge is moving around in a circle at a linear velocity of (ω * r) meters per second.
1 amp of current = 1 coulomb per second
Amps = coulomb/meter * meters/sec
Amps = (σ * 2 * π * r) * (ω * r)
So, the current flowing on the inner surface of the cylinder = (σ * 2 * π * r) * (ω * r) amps
This amount of current flowing in a circle on the inner surface of the cylinder produces a magnetic field. This magnetic field induces a current in the outer surface of the shell.
The strength of the β field is inversely proportional to the square of the distance between the source of the β field and the inner surface of the shell. The distance = r + δ. Since δ<<r , we can say that the distance = r
So, the β field at the outer surface of the shell = μo * (σ * 2 * π * r) * (ω * r) ÷ r^2
r * r ÷ r^2 = 1
So, the β field at the outer surface of the shell = μo * (σ * 2 * π) * (ω)
This β field will induce a voltage in the outer surface of the shell, which is a conductor with resistivity ρ and thickness δ
The induced voltage = β outer * (length) * (velocity)
(length) * (velocity) = area per second.
Induced voltage = β outer * area per second.
Area per second is the distance that the area of the inner surface moves each second.
The area of the inner surface = 2 * π * r * length
Linear velocity of cylinder = (ω * r) meters per second.
This means the outer surface of the cylinder is moving (ω * r) meters per second in a circular path.
This means that a surface with area of (2 * π * r) m^2 is moving (ω * r) m/s
Area per second = (2 * π * r) * (ω * r)
We let the length of the cylinder = 1 meter
Area per second = (2 * π * r) * (ω * r)
β outer = μo * (σ * 2 * π) * (ω)
Induced voltage = μo * (σ * 2 * π) * (ω) * (2 * π * r) * (ω * r)
V = I * R
Resistance = (resistivity * length) ÷ area
Length = 1
Resistance = ρ ÷ (2 * π * r)
Current = V ÷ R
Current = μo * (σ * 2 * π) * (ω) * (2 * π * r) * (ω * r) ÷ ρ ÷ (2 * π * r)
angular acceleration = τ ÷ [mr^2 + (1/ ρ * π * δ * l * μo^2 * σ^2 * r^2 * ω)
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Sagar Singh
B.Tech IIT Delhi
