# When to consider motional emf and when not to?

In this configuration(given in picture) ,the magnetic flux through the loop is changing(which causes an induced electric field,which causes an induced emf) and for the perpendicular rods AB and CD(if we consider them to be not a part of the loop and moving with velocity v on their own(just for visualization) then shouldn't we consider motional emf for them and add both induced and motional emfs to give resultant emf in loop? I was told that I could do only one of them at a time not both.The explanation given to me was not satisfactory please help

the configuration:

• If I understand what you're saying correctly. If the loop is moving. Then the total instantaneous emf in the Loop is the sums of the line integral of all of the sides, (provided that we perform the line integrals connected with eachother, ie, (A->D) + (D->C) + (C->B) + (B->A). What do you mean " add induced and notional emfs"? there are only induced motional emfs here. Jan 27, 2022 at 11:11
• @jensenpaull Doesn't faraday law state that if in a loop magnetic flux changes then there is an induced emf in the loop?(acc to what my instructor said this will cause an induced electric field(non conservative one)but acc to some books which I read, that happens only if there is time varing magnetic field does it. And yes you understood me correctly. Jan 27, 2022 at 11:23
• What is correct Jan 27, 2022 at 11:25
• Acc to wiki pedia any change in magnetic flux causes an induced emf:en.wikipedia.org/wiki/Faraday%27s_law_of_induction in the mathematical description section which makes we think why not add that induced emf with motional emf too?? Jan 27, 2022 at 11:39
• In your given scenario, no transformer emf is present. There is no electric field. This doesn't mean there can't be an emf Jan 27, 2022 at 12:17

## 1 Answer

Comment answer:

There is a subtle difference in Faradays law compared to the maxwell-faraday quation( you may not have known there are 2 to begin with)

Faraday law:

$$\epsilon = -\frac{d\phi_{B}}{dt}$$

$$\epsilon = -\frac{d}{dt}\int \vec{B} \cdot \vec{da}$$

describes both motional emf and transformer emf, ANY change in magnetic flux on a surface causes an EMF. this doesn't necessarily say whether or not the EMF is caused by an induced electric field, or if it is caused by the magnetic lorentz force. This equation COULD give rise to an emf( changing magnetic flux), due to a change in the boundary of the surface in question, or due to a changing B field($$\frac{\partial \vec{B}}{\partial t})$$ within that boundary

Whereas the maxwell-faraday equation $$\int \vec{E} \cdot \vec{dl} = -\int \frac{\partial \vec{B}}{\partial t} \cdot \vec{da}$$

only describes transformer emf, aka an emf caused by an induced electric field as a result of a changing magnetic field

In this situation there is only motional emf, as there is a NON changing magnetic field as its the field of a static wire. But, there is a moving loop, aka a changing magnetic flux caused by a change in the boundary of my surface. In this situation there is NO induced electric field. The EMF generated in this situation is not described by the maxwell-faraday equation. It is instead described by faradays law. With the physical mechanism for this EMF being the magnetic lorentz force on the charges in the boundary of my surface.

To find the emf I could compute the changing magnetic flux in my loop using FARADAYS law, to find the EMF. Or... as you have suggested. I Could use the lorentz force to calculate seperately the line integrals of all of the sides to find the total emf in the loop

Where

$$\epsilon_{i} = \int_{a}^{b} (\vec{V}×\vec{B} ) \cdot \vec{dl}_{i}$$

To mathematically understand why there is a difference between faradays law and the maxwell faradays equation. Is that the time derivative is in the inside of the integral. This forces the equation to completely ignore any change to the surface boundary

• brilliantly cleared now. Thankyou so so much for this god bless you bro. So that means an induced emf(transformer one) comes into picture only when magnetic field is time varying.. otherwise its always going to be due to motional emf right or changes in the shape of my loop Jan 27, 2022 at 12:46
• Yup! This is also seen from the differential form that $$\nabla × \vec{E} = - \frac{\partial\vec{B}}{\partial t}$$if $\frac{\partial\vec{B}}{\partial t}$ is zero, then the induced electric field must also be zero, and thus any emf is caused by the magnetic lorentz force on charges in my wire! Jan 27, 2022 at 12:49
• Look at the derivation of the motional emf from that same Wikipedia page it explains more clearly how we can explicitly extract motional emf from faradays law! Jan 27, 2022 at 12:53
• ok thankyou so much. Jan 27, 2022 at 12:53
• (^^ technically emfs can be also be caused by static E fields actually, not just B fields, when no changing B fields are present)** Jan 27, 2022 at 12:55