Balancing area of a loop and magnetic field through it

[uzair_ha91]

Question: Is it possible to change both the area of the loop and the magnetic field passing through the loop and still not have an induced emf in the loop?
Equation: change in Magnetic flux= area * magnetic flux density
My attempt: If the magnitude of magnetic field is changing, then by either increasing or decreasing the area of the loop, the change in magnetic flux associated with the loop can be kept constant. Then the induced emf will be zero.
Can you somehow prove this through the equation, numbers? Would appreciate the help.

 

[rock.freak667]

Φ=BA


Say initially you have Φ₀=B₀A₀. Then decrease B0 by 1/2 (so B₁=1/2B₀) and you increase the area by a factor of 2 (A₁=2A₀)

Φ₁=B₁A₁ =(1/2 B₀ *2A₀)=B₀A₀=Φ₀

if the changes occur in the same time interval,t, then E₀=E₁

Once there is a change in magnetic flux, there will be an emf induced. So as much as I can say, even if you decrease one and increase the other and emf is still being induced, it just has the same value as before. So the net emf induced would be zero.

 

[tomcue]

Yes, it is possible to change both the area of the loop and the magnetic field passing through it and still not have an induced emf in the loop. This is because the induced emf is dependent on the change in magnetic flux, which is equal to the product of the area and the magnetic flux density.

If the magnetic flux density remains constant, then changing the area of the loop will result in a change in magnetic flux. However, if the change in the area is proportional to the change in the magnetic field, then the change in magnetic flux will cancel out and result in no induced emf in the loop.

To prove this, we can use the equation given: change in magnetic flux = area * magnetic flux density. Let's assume that the initial area of the loop is A and the initial magnetic field passing through it is B. The initial magnetic flux is then given by Φ = A*B.

Now, let's say we change the area of the loop to A' and the magnetic field to B'. The new magnetic flux is given by Φ' = A'*B'. If we substitute this in the equation for change in magnetic flux, we get:

Change in magnetic flux = Φ' - Φ = A'*B' - A*B

Since we want the change in magnetic flux to be zero, we can set this equation equal to zero and solve for A' in terms of B'. This gives us A' = (A*B)/B' = A*(B/B'). This means that for every change in the magnetic field, the area of the loop must change in proportion to maintain a constant change in magnetic flux.

For example, if we double the magnetic field, we must also double the area of the loop to maintain a constant change in magnetic flux. This will result in no induced emf in the loop.

In conclusion, it is possible to change both the area of the loop and the magnetic field passing through it and still not have an induced emf in the loop as long as the change in area is proportional to the change in magnetic field. This can be proven through the equation and can also be demonstrated with specific numerical values.