# Example problem of induced emf in a coil travelling with some velocity and perpendicular to the earth's magnetic field

I have a example problem which I created myself just out of curiosity,

Question:

Let's say a copper wire coil(20mm dimeter(r=10mm)) of 3 turns(N=3) is moving along the surface(Note-coil is not rotating) of the earth from east to west at the speed of 300 km/sec(|v|=300 km/sec);The magnetic field strength of the earth(|B|=60E-6 T) are passing through the coil(through the hole of the coil) perpendicular to the velocity vector of the coil(just for this example ignore the fact that magnetic field is 15 degrees tilted from the geographical pole north); How much current/emf will be induced in the coil?

I] According to the faraday's law of induction, the emf will only be induced when flux is changing with respect to time, otherwise emf will be zero. so for the above problem, the answer is emf=0,

$$Induced \space emf = -N \times \frac{d(\phi)}{dt} volts$$

II] According to the vector formula for induced emf,

Explanation for below formula for induced emf in a wire placed near a constant magnetic field.

$$Induced \space emf = L \times (v \times B) volts$$

where v is he velocity vector and L is the total length of the coil's wire and B is magnetic field.

If we take this formula and calculate the answer we are sure get a value >0 for induced emf and induced current.

So, what I am asking is which one is the correct answer? Or I made a mistake somewhere?

You are calculating for two different scenarios. I] is a coil, II] is a single wire. If you use a wire forwards and another wire back, to make the two sides of the coil perpendicular to the field, they add in anti-phase to give you zero.

• So, the II] vector form for induced emf can be only used for straight wires or plane conductors? Is it not applicable for a coil? I considered a coil for both scenarios. Commented Mar 26 at 17:54
• @yssh - The idea Neil expressed is that a coil (or loop) can be thought of as two parallel wires connected end to end. When both wires are moving together across the same plane of a magnetic field there could be charge motion in both wires, but the direction of the charge will be the same for both wires, resulting in no net current around the loop.
– Nedd
Commented Mar 27 at 6:21
• @yssh No, you did not consider a loop for both scenarios. You used a coil for 1, and a single wire for 2. Commented Mar 27 at 8:22
• @Neil_UK I understood, But is it possible to consider a loop of wire with the 2nd equation? and if so how the 2nd equation will look like? Commented Mar 27 at 11:06
• @yssh Of course it's possible to consider a loop of wire with the 2nd equation, that's what my answer does. It's not what your question does. Commented Mar 27 at 11:45