Hi
Please note the following
- Electric field, E = force per unit charge
- Voltage, V = work done per unit charge (in moving it by a distance, d, in an electric field E)
- So V = E*d
- Lorentz law: magnitude of Lorentz force, F, on an electric charge, q, in an electric field, E = q*u*B (B=magnetic field, u=velocity of the charged particle)
- Direction of Lorentz force will be obtained by cross product of vectors u and B, in that order.
w = angular velocity of the conducting wheel in clock wise direction
B = magnetic field intensity pointing out of the screen of the this computer / mobile / tablet screen
R = length of the each of the radial rods
r = radial distance of each of the point on the rod from center of the circle
n = number of such rods connecting the periphery and center of the wheel
e = magnitude of the charge of an electron
u = velocity of the rod at a radial distance r from center
Draw a circle and indicate the w in clock direction and B with ‘.’ marks, indicating that this is acting out of the screen. From this figure, it will be clear that each of the point on the rod will have a tangential velocity equals to r*w. Let the conducting rods carry some +ve charged particles.
From Lorentz law, force on +ve charges inside the conducting rodis F=q*u*B and the direction will be towards center. So these +ve charges get accumulated at the center of wheel (for the given configuration of B). Note: If B is acting into the scree, +ve charge accumulation will be on the periphery of wheel.
We know, E = F / q = u*B = r*w*B (since u=r*w)
Let dV is the infinitesimal potential difference created by this force E (force per unit charge) in moving an unit +ve charge through a distance dr along the rod towards center. And that will be equal to E*dr.
so dV = r*w*B*dr.
Now to get the total induced potential difference in one rod, the above equation has to be integrated between limits r=0 and r=R. After integration we get
V=(w*B*R^2)/2
This is the work that is done in bringing an unit +ve charge from the periphery to center of the wheel in one rod. So, if there are n identical rods, the total work done per unit charge (total induced potential difference) will be n times V.
For the given problem, n=4 nos, w=2 rad / sec, B=8 Tesla, R=1/4 metres;
So total induced potential difference for 4 rods = 4*(2*8*0.25^2)/2 = 2 Volts (Answer)
All the best.
