A metal disc of radius R rotates with angular velocity Omega (w) about an axis perpendicular to its plane passing through its centre in a magnetic field B acting perpendicular to the plane of the disc calculate the induced EMF between the rim and the axis of the disc
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To calculate the induced EMF (electromotive force) between the rim and the axis of the disc, we can use Faraday's law of electromagnetic induction. According to this law, the induced EMF is equal to the rate of change of magnetic flux through the area enclosed by the conducting loop.
In this case, the conducting loop is the metal disc, which has a radius R. The magnetic field B is acting perpendicular to the plane of the disc. The magnetic flux through the disc is given by the equation:
Φ = B * A
where Φ is the magnetic flux, B is the magnetic field, and A is the area of the disc.
The area of the disc can be calculated using the formula:
A = π * R^2
Now, let's consider the disc rotating with angular velocity ω. As the disc rotates, the area enclosed by the conducting loop changes with time. The rate of change of the area is given by:
dA/dt = 2πR * (dR/dt)
Since the disc is rotating, the rate of change of the radius with respect to time is equal to the tangential velocity at the rim, which is given by:
v = R * ω
Now, we can substitute the values in the equation for the rate of change of area:
dA/dt = 2πR * (dR/dt) = 2πR * (v/R) = 2πv
Therefore, the rate of change of the area is 2πv.
Finally, we can calculate the induced EMF using Faraday's law:
EMF = -dΦ/dt = -B * dA/dt = -B * 2πv
So, the induced EMF between the rim and the axis of the disc is equal to -B * 2πv.
To calculate the induced electromotive force (EMF) between the rim and the axis of the disc, we can use Faraday's law of electromagnetic induction. According to Faraday's law, the induced EMF is equal to the rate of change of magnetic flux through the disc.
The magnetic flux through the disc can be calculated using the formula:
Φ = B*A
Where:
Φ is the magnetic flux
B is the magnetic field strength
A is the area of the disc
In this case, the magnetic field acts perpendicular to the plane of the disc, so the area of the disc is given by:
A = π*R^2
Where:
R is the radius of the disc
Now, let's consider the rotation of the disc. As the disc rotates, the area vector associated with the disc changes with time. The rate of change of the area vector is given by:
dA/dt = -ωA
Where:
dA/dt is the rate of change of the area vector
ω is the angular velocity of the disc
Substituting the expression for the area A, we get:
dA/dt = -ωπR^2
Finally, substituting the expression for the rate of change of the area vector into Faraday's law, we get:
EMF = -dΦ/dt = -B * dA/dt = ωπR^2B
So, the induced EMF between the rim and the axis of the disc is given by:
EMF = ωπR^2B
Note that this calculation assumes a uniform magnetic field and neglects any effects of resistance or other factors that might affect the actual induced EMF in a real scenario.
speed of any length dr on the disk=Omega*r
E= -BA/t so applying here
B given
dA/dt= dr*dAngle/dt
=dr*rw
E=-B wr dr
Etotal=INT Bwrdr over r=0 to R
Etotal=BWR^2/2
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