Magnetic Flux

Magnetic flux is the surface integral of the magnetic field normal to a given area. In simpler terms without the use of multivariate calculus, magnetic flux is the dot product of the magnetic field and a given area, expressed as

If this definition doesn't make sense, think of it like this. In physics, flux is just the rate of flow so keep that in mind. This equation tells us that the rate of flow of total magnetic field line vectors going through a given area is proportional to the magnitude(amount) of those field lines and the area they go through. If the area is greater, more field lines would go through so more magnetic flux would be generated. If the magnetic field itself is greater, that too will allow for more magnetic flux. However, there's one component that we're missing. If the magnetic field were perpendicular to the area, no magnetic field lines would go through the given area, meaning that there would be no magnetic flux. This means that we have to make a dot product for the flux to make sure that we factor in the angle between the magnetic field and the area. The units for magnetic flux are in Webers(Wb).

So, we've established that currents are what drives magnetic fields to work, whether on a macroscopic or microscopic scale. However, currents themselves have major applications in circuit analysis and electronics, namely in circuits themselves. Thus, it would be apt to combine the concepts of circuits and magnetic fields under a specific set of phenomena. Well, that's would Faraday's Law of Electromagnetic Induction does.

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Faraday's law states that the induced electromotive force in a closed path is equal to the negative of the rate of change of the magnetic flux enclosed by the path with respect to time.

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If that was a mouthful, let's break it down both conceptually and mathematically.

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Conceptually, the statement tells us that if we take a loop enclosed by a circuit and change the magnetic flux going through that loop, we'll produce an emf in the circuit.

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Mathematically, the statement translates to this

The term with N is for if the loop in question has multiple, or N, turns, like a solenoid. Otherwise, if the loop has 1 turn, N = 1. The negative sign is in there to account for the fact that the induced emf is equal to the negative rate of change of magnetic flux with respect to time, which is represented in the equation, too. Note that the fraction on the rightmost term of the equation can be expressed as a derivative for infinitesimal intervals of time.

Lenz's law is extended from Faraday's Law and states the current and magnetic field created by the induced emf due to a magnetic flux generates a magnetic field which opposes the effect which produced it.

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Basically, if we change the magnetic flux through a loop and induce an emf, that emf will produce a current which, in turn, will generate a magnetic field. The generated magnetic field and current will act such that they oppose the change in magnetic flux. If the magnetic flux doesn't change, of course, no emf, and thus no current is produced. If the magnetic flux increases, the current will go a certain direction, and if the magnetic flux decreases, the current will go the opposite direction. In this way, the total magnetic field is constant as the opposing magnetic fields neutralize.

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This phenomenon usually gets represented by circuit loops with magnetic fluxes going through them. To determine the direction of current you can use the second right-hand rule, where you make a thumbs-up sign with your right hand. The process for determining can be extremely confusing at first but it just takes a bit of getting used to.

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Let your thumb point in the "opposite" direction of the magnetic field lines. For example, if the magnetic field lines were directed through the loop going up, your thumb would represent the downward direction.

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The curl of your fingers represents the direction in which the current is flowing on the side you're inspecting from. In the same example, if your thumb's direction was "down", then the curl of your fingers, which is the direction of the current, is clockwise if you were to look at the loop from above going down. If your thumb's direction was "up", then the curl of your fingers, which is the direction of the current, is clockwise if you were to look at the loop from below. However, a clockwise current from below is actually counterclockwise from above, and that makes the distinction.

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You're almost done but there's one final but extremely important step. Assuming you've completed the prior steps, you've established the direction of current flow relative to the perspective given by your thumb. Now, you must determine if the magnetic flux is increasing or decreasing.

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If the magnetic flux decreases, you have to switch the direction you established for the current flow relative to the direction given by your thumb. If the direction was established as clockwise, it would now be counterclockwise in that same perspective, and vice versa. If the magnetic flux increases, then you don't have to switch the direction. Of course, if the magnetic flux is constant, then you don't have to worry about any of this.