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I am studying magnetic flux linkage in an ac generator and it appears to be that magnetic flux linkage is negative half the time, how can this be?? Also with lenz's law why is emf defined as negative when magnetic flux is increasing and how does this relate to the direction of the current?

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3 Answers 3

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Yes, magnetic flux can be negative. It just depends on where the field is going. Say there is a sheet and magnetic field is going through it from front to the back, we can call the flux there as positive and negative when it's the other way round.

It is pretty clear from the statement of Lenz's Law why the emf defined is taken as negative: An induced electromotive force (emf) always gives rise to a current whose magnetic field opposes the original change in magnetic flux. (wikipedia)

Basically, the magnetic field produced due to the induced current opposes the magnetic flux producing the current itself.

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But why could this emf not be positive and the emf in the other direction be negative, what defines the direction of the emf?? –  Joseph Apr 5 at 10:18
    
It can be the other way round too, would not make a difference. The sign just shows the direction, the magnitude is what matters. –  Parth Vader Apr 5 at 10:34
1  
How do we know what direction relates to what sign?? –  Joseph Apr 5 at 10:37
    
We can assume any direction to be positive. The other side would be negative. –  Parth Vader Apr 5 at 11:05
    
So why do we need the minus sign in faraday's law? if emf can be positive when flux is increasing? –  Joseph Apr 5 at 11:14

Magnetic flux is a scalar quantity and its positive/negative sign indicates the direction of the magnetic field.

And the Faraday's law of induction is a quantitative version of Lenz's law, which may help your understanding: $\oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} = - \frac{d}{dt} \iint_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{S}$

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magnetic flux linkage is negative

A good observation. Now, the magnetic flux associated with a $surface$ is given by $\iint{ }{ }\vec{B}\cdot\hat{n}\mathrm{d}A$

The surface could be an open one or a closed one. Let us consider an open surface as shown below

enter image description here

For an open surface, the area vector could be in any direction ( I've chosen an arbitrary direction which in this case is towards east ). $\hat{n}dA$ in the equation of flux tells us that the area vector is always normal to the surface.

enter image description here

In the above figure, a uniform magnetic field exists in space. The flux associated with the surface can be calculated as follows

$$\iint{ }{ }\vec{B}\cdot\hat{n}\mathrm{d}A = BA$$

This is because $\vec{B}$ and $\hat{n}\mathrm{d}A$ are in the same direction and hence the dot product is $1$. Also, $\vec{B}$ is uniform and can be taken out as a constant from the integral. So, we get a positive flux.

Now, what if things looked like this

enter image description here

In this case you observe that $\vec{B}$ and $\hat{n}\mathrm{d}A$ are in the opposite directions which tells us that the dot product will be $-1$. In this case, the flux is negative.

Mathematically,

$$\iint{ }{ }\vec{B}\cdot\hat{n}\mathrm{d}A = -BA$$

So, this is how you calculate the flux for an open surface. I've stressed on an open surface alone since we do not use a closed surface to calculate the flux while working with generators. We come across closed surfaces while dealing with Gauss's law where the closed surface is also called the Gaussian surface.

Also with lenz's law why is emf defined as negative when magnetic flux is increasing and how does this relate to the direction of the current?

From experiments it was observed that a change in the magnetic flux( associated with a current carrying loop ) induced a current in the loop whose magnetic field opposed the change in magnetic flux.

enter image description here

So, if you move a bar magnet towards a conducting loop such that the magnetic flux associated with the loop surface increases, there will be an induced current( whose direction is given by the red arrow ) whose magnetic field opposes the increase in magnetic flux. From the figure above, you clearly see that the induced current creates a magnetic field in a direction opposite to the external magnetic field created by the bar magnet.

Look at this figure

enter image description here

If you move the magnet away from the conducting loop, there will be a decrease in the magnetic flux associated with the loop surface. So now the induced current will flow in a direction so as to oppose this decrease in flux. You can clearly see that the direction of the magnetic field created by the induced current is in the direction of the external magnetic field.

This is the prime reason behind why there is a negative sign in Faraday's law. The negative sign is explained by Lenz's law. It tells us that the induced E.M.F always opposes the change in magnetic flux associated with a conducting loop.

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