When you hear the words magnetic flux, your first thought might jump to Back to the Future and Doc Brown's De-Lorean powered by the flux capacitor. Unfortunately in reality, magnetic flux doesn't appear to allow one to travel time no matter how cool your car. However in physics, magnetic flux is an incredibly important concept which is central to the idea of electromagnetic induction, something almost all of our electric power relies on. In this article, we'll dig deeper into magnetic flux and how one can calculate it. We'll also see how a changing magnetic flux can induce a current thanks to Faraday's Law.
Magnetic Flux Definition
Let's start with the basic definition of what exactly magnetic flux is.
Magnetic Flux is a measurement of the total amount of magnetic field passing through a given surface area.
From this definition we can see that the magnetic flux is dependent on two quantities, the magnetic field \(\vec{B}\) and the surface area \(\vec{A}\) of the surface we are interested in. This surface area is expressed as a vector \(\vec{A}=A\vec{n}\) where \(A\) is the magnitude of the surface area and \(\vec{n}\) is a unit vector perpendicular to the surface, known as the normal vector. For example, in Cartesian Coordinates, the normal vector for a horizontal surface lying in the \(x-y\) plane is the \(\vec{k}\) unit vector parallel to the \(z\) axis. For a sphere, the normal vector is a unit vector \(\vec{r}\) parallel to the radius of the sphere.
Unit vectors are vectors of length one, in Cartesian coordinates we use the notation \(\vec{i},\,\vec{j},\,\vec{k}\) to denote the unit vectors in the \(x,\,y,\,z\) directions respectively.
Note that these surfaces do not need to be real physical surfaces, we often want to consider the flux through an imaginary mathematical surface when making calculations. Most of the time, this allows us to use surfaces that simplify some integral calculations.
When trying to visualize magnetic fields we often utilize the concept of field Lines, which are imaginary lines, tangent to the force vectors of the field, representing the direction and magnitude of the force experienced by a test charge in the magnetic field. We can define magnetic flux as the net number of field lines passing through a particular surface.
Fig. 1 - Magnetic flux can be represented with flux or field lines which show the direction of force. The more field lines pass through a surface the greater the flux will be.
It's important to consider the direction of the field lines; if the surface has an equal number of field lines entering as leaving it the net flux will be zero.
Magnetic Flux Equation
We can turn this intuitive definition of magnetic flux into a precise mathematical definition using the following equations. There are two main situations we need to consider, one in which the field has a constant value at all points on the surface and one in which the field varies across the surface.
Constant Field
If the field, \(\vec{B}\), is the same across the entire surface, \(\vec{A}\), then the following equation can be used for the magnetic flux \(\Phi_B\):
\[\Phi_B=\vec{B}\cdot\vec{A}.\]
Using the definition of the dot product, we see that the magnetic flux is equal to the component of magnetic perpendicular to the surface multiplied by the surface area, as discussed in the previous section. If the angle between the normal vector and the magnetic field is known, the magnitude of the magnetic flux can be expressed as:\[\begin{align}\Phi_B&=\vec{B}\cdot A\vec{n},\\&=|B|A\cos(\theta),\end{align}\]
where \(\theta\) is the angle between the field vector and the surface normal vector, \(|B|\) is the magnitude of the magnetic field. The magnetic field strength is measured in units of Teslas \(\mathrm{T}\), meaning that the magnetic flux is measured in units of Tesla-meters squared \(\mathrm{T}\mathrm{m}^2\) also known as Weber's \(\mathrm{Wb}\).
Consider a uniform magnetic field \(\vec{B}=6\vec{i}+3\vec{j}\,\mathrm{T}\) passing through a square region in the \(x-z\) plane, with sides of length \(2\,\mathrm{m}\), whose normal vector is \(\vec{n}=\vec{j}\). What is the magnetic flux through this surface?
First note that the surface area of the square is \(4\,\mathrm{m}^2\) and the area vector is given by \(\vec{A}=4\,\mathrm{j}\,\mathrm{m}^2\).
The formula for magnetic flux shows us that we need to take the dot product of the magnetic field and the area vector\[\begin{align}\Phi_B&=\vec{B}\cdot\vec{A}\\&=\left(6\vec{i}+3\vec{j}\,\mathrm{T}\right)\cdot\left(4\,\vec{j}\,\mathrm{m}^2\right)\\&=12\,\mathrm{Wb}.\end{align}\]
Varying Field
If instead the field varies over the surface we are considering, things become a little trickier. Here, we have to use calculus. The idea is that we consider the amount of field \(\vec{B}\left(\vec{r}\right)\) flowing through an infinitesimal piece of the surface \(\mathrm{d}\vec{A}\) by taking the dot product \(\vec{B}\left(\vec{r}\right)\cdot \mathrm{d}\vec{A}\),
and then we integrate over each infinitesimal piece of surface to find the total flux through the entire surface
\[\Phi_B=\int_S\vec{B}\left(\vec{r}\right)\cdot\mathrm{d}\vec{A}.\]
The integral symbol \(\int_S\) denotes a surface integral.
The exact calculation of the surface integral depends massively on the type of surface we are investigating, in general it is only simple surfaces such as squares or spheres that we can easily calculate exactly. Let's take a look at an example to see how this works.
Consider a magnetic field that varies over the surface of a square with sides of length \(2\,\mathrm{m}\) lying in the \(x-y\) plane. The magnetic field is described by the function\[\vec{B}\left(\vec{r}\right)=|B|\left(\frac{1}{x^2}+\frac{1}{y^2}\right)\vec{k}.\]
To find the magnetic flux, first note that the infinitesimal area vector of the square is given by
\[\mathrm{d}A=\mathrm{d}x\mathrm{d}y\vec{k}.\]
If we choose the coordinate system so that the corners of the square are at \((x,y)=(\pm 1, \pm 1),(\pm 1, \mp 1)\). The integral for the flux then looks like this.
\[\begin{align}\Phi_B&=\int_{S} |B|\left(\frac{1}{x^2}+\frac{1}{y^2}\right)\vec{k}\cdot\left(\mathrm{d}x\mathrm{d}y\right)\,\vec{k}\\&=\int_S |B|\left(\frac{1}{x^2}+\frac{1}{y^2}\right)\mathrm{d}x\mathrm{d}y\\&=\int_S |B|\left(\frac{\mathrm{d}x}{x^2}+\frac{\mathrm{d}y}{y^2}\right)\\&=|B|\left(\int_{-1}^1\frac{\mathrm{d}x}{x^2}+\int_{-1}^1\frac{\mathrm{d}y}{y^2}\right)\\&=|B|\left(\left[-\frac{1}{x}\right]_{-1}^1+\left[-\frac{1}{y}\right]_{-1}^1\right)\\&=|B|\left(2+2\right)=-4|B|\,\mathrm{Wb}.\end{align}\]
Magnetic Field vs Magnetic Flux
Whilst the two concepts are intimately related, it's important that we don't confuse the concept of magnetic flux with magnetic field. The most essential thing to remember is that magnetic flux is also determined by the surface area of a given surface, whereas magnetic field simply indicates the force felt by a charge at a given point. This means that for a single magnetic field, a whole range of possible fluxes can arise depending on the surface we are considering and its position in relation to the field.
What's more, the magnetic flux is determined by the net amount field, which is determined by the direction of each field line passing through the surface. This means that it is possible for a field which is non-zero everywhere to produce zero flux, if the amount of field flowing into a surface is equal to the amount of field flowing out. In fact, Gauss' Law for magnetic fields states that for any closed surfaces the total flux through the surface is always zero. This is due to the fact that magnetic monopoles cannot exist within nature.
Gauss' Law for magnetic fields states that the total flux through a closed surface, containing no holes, is always equal to zero.
Mathematically this is given as\[\oint_S\vec{B}\cdot\mathrm{d}\vec{A}=0.\]
So we see that magnetic field and magnetic flux are in fact distinct properties that can tell us very different things about the strength of a magnetic field.
Change in Magnetic Flux
So we've got to grips with what magnetic flux is and how we can calculate it, but why exactly is magnetic flux such an important quantity in physics? The answer lies in the phenomena of electromagnetic induction.
Electromagnetic induction refers to the process of creating electromotive forces (EMF), which can produce currents, by moving a magnetic field around an electric conductor or by moving an electrical conductor through a fixed magnetic field.
Electromagnetic induction was first discovered and analyzed by Michael Faraday in his experiments on electromagnetism in the 1830s. Faraday's great discovery was that when two coils of wire were wrapped on either side of an iron bar, passing a current through one coil would momentarily induce a current in the other coil.
Off the back of these experiments, Faraday postulated his law of electromagnetic induction.
Faraday's Law of Induction - 'The EMF around a closed path is equal to the negative of the time rate of change of the magnetic flux enclosed by the path'
So we see that it is a changing magnetic flux that directly determines the magnitude of the induced EMF. So in the case of the coils and iron bar, when the current is first switched on the change in EMF induces a magnetic field in the iron bar. This sudden change in the flux in the magnetic bar induces an EMF and current through the other coil.
Mathematically this statement is one of Maxwell's equations, the fundamental laws of classical electromagnetism. First, note that the EMF \(\mathcal{E}\) around a closed loop can be given as closed line integral of the electric field:\[\mathcal{E}=\oint_{\partial S}\vec{E}\cdot\mathrm{d}\vec{l}\]
\(\partial S\) denotes the edge of a surface \(S\) so for example if \(S\) is a circle \(\delta S\) is a closed loop.
A line integral is similar to a surface integral, in that we integrate over the electric field at each infinitesimal segment of the line. The difference is here we are interested in the components of field parallel along the line rather than perpendicular as in the surface integral. We also only have to integrate over one dimension. The formula for Faraday's Law is then:\[\begin{align}\mathcal{E}&=-\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}\\\implies \oint_{\partial S}\vec{E}\cdot\mathrm{d}\vec{l}&=-\frac{\mathrm{d}}{\mathrm{d}t}\int_S\vec{B}\left(\vec{r}\right)\cdot\mathrm{d}\vec{A}\end{align}\]
It's important to remember that because Faraday's law concerns a changing magnetic flux, there are two possible ways to induce an EMF in a coil. The first is obviously to use a variable magnetic field source and change the strength or direction of the field. The second is to change the amount of surface area exposed to a fixed magnetic field. This second method is often a much simpler method of inducing EMFs. For example, in wind turbines, a magnet, held between coils of wire, is rotated by the spinning blades. As the amount of coil exposed to the field is constantly changing as the magnet rotates this also produces a changing flux and hence an EMF.
Magnetic Flux Problems
Let's look at some example problems regarding magnetic flux.
Q: Consider the magnetic field produced by a long thing piece of current carrying wire given by \(\vec{B}=\frac{\mu_0I}{2\pi r}\vec{\theta}\) where \(\mu_0=4\pi\times10^{-7}\,\mathrm{N}\,\mathrm{A}^{-2}\) is the permeability of free space and \(r\) is the radial distance from the wire. The unit vector \(\vec{\theta}\) describes the fact that the magnetic field curls around the wire.
A: Give an expression for the flux through a circular surface of radius \(R\) which lies in the plane of the wire, such that the magnetic field lines are perpendicular to the surface.
Let's start with the integral formula for magnetic flux \[\Phi_B=\int_S\vec{B}\left(\vec{r}\right)\cdot\mathrm{d}\vec{A}.\]
The differential surface area is given by\[\mathrm{d}\vec{A}=r\mathrm{d}r\,\mathrm{d}\theta\vec{\theta}\]
Plugging this in with the magnetic field equation given in the question we find:\[\begin{align}\Phi_B&=\int_S\vec{B}\left(\vec{r}\right)\cdot\mathrm{d}\vec{A}\\&=\int_0^R\mathrm{d}r\int_0^{2\pi}\mathrm{d}\theta\left(\frac{\mu_0I}{2\pi r}\right)\vec{\theta}\cdot r\vec{\theta}\end{align}\]
The vectors are clearly parallel so their dot product is \(1\). As the magnetic field does not depend on \(\theta\) we can integrate over \(\theta\) immediately to get \(2\pi\).\[\begin{align}\Phi_B&=2\pi\int_0^Rr\mathrm{d}r\left(\frac{\mu_0I}{2\pi r}\right)\\&=2\pi\int_0^R\mathrm{d}r\left(\frac{\mu_0I}{2\pi}\right)\\&=2\pi R\left(\frac{\mu_0I}{2\pi}\right)\\&=R\mu_0I\end{align}\]
So we see that the amount of flux passing through the circular surface is dependent only on the current and the radius of the surface.
Q: Consider a time dependent magnetic field defined by the function \(\vec{B}(t)=B\sin\left(2\pi t\right)\vec{z}\). If a circular loop of radius \(r=0.1\,\mathrm{m}\) is placed in the field such that its radial vector \(\vec{r}\) is at an angle of \(\theta=45^{\circ}\,\mathrm{deg}\) with the magnetic field direction \(\vec{z}\). What will the value of the induced EMF \(\mathcal{E}\) be?
A: Faraday's law tells us that the EMF induced by an oscillating magnetic field is proportional to the rate of change of the magnetic flux.
\[\mathcal{E}=-\frac{\mathrm{d}}{\mathrm{d}t}\int_{S}\vec{B}\cdot\mathrm{d}\vec{A}\]
Let's first find an expression for the magnetic flux.
\[\begin{align}\Phi_B&=\int_{S}\vec{B}\cdot\mathrm{d}\vec{A}\\&=\int_SB\sin\left(2\pi t\right)\vec{z}\cdot\vec{A}\end{align}\]From the definition of the dot product and the angle given in the question we know\[\vec{z}\cdot\mathrm{d}\vec{A}=\cos\left(45\right)\mathrm{d}A=\frac{\sqrt{2}}{2}\]Note that the magnetic field is spatially independent and so we can take it outside of the integrand. \[\Phi_B=\frac{\sqrt{2}}{2}B\sin\left(2\pi t\right)\int_{S}\mathrm{d}\vec{A}\]The integrand now just gives the surface area enclosed by the circular loop, which is \(\pi r^2=\frac{\pi}{100}\).\[\Phi_B=\frac{\sqrt{2}\pi}{200}B\sin\left(2\pi t\right)\]To find the EMF, we need to take the derivative with respect to time \[\mathcal{E}=\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}=\frac{\sqrt{2}\pi^2}{100}B\cos\left(2\pi t\right)\]
We can consider one more example to strengthen our understanding.
Q: Consider a solenoid producing a fixed magnetic field \(B=10\,\mathrm{T}\). If a loop of wire of radius \(R=0.5\,\mathrm{mm}\) is rotated such that the angle between the normal vector and the magnetic field is given by \(\theta=2\pi t\). Determine an expression for the EMF induced in the loop, and find the value of the EMF after \(0.5\,\mathrm{s}\)
A: As the magnetic field is fixed, we can use the fixed field equation for magnetic flux.
\[\Phi_B=|B|A\cos(\theta)\]
The surface area \(A\) is given by \(\pi R^2=\frac{\pi}{4}\,\mathrm{mm}^2\)
So we see that the time-dependent magnetic flux is given by\[\Phi_B(t)=|B|A\cos(2\pi t)\]
Faraday's Law then tells us that the EMF induced in the loop is given by\[\begin{align}\mathcal{E}&=-\frac{\mathrm{d}\Phi_B(t)}{\mathrm{d}t}\\&=|B|A2\pi\sin(2\pi t)\\&=10\,\mathrm{T}\cdot\frac{\pi}{4}\,\mathrm{mm}^2\cdot2\pi\cdot\sin(2\pi t)\\&=5\pi^2\sin(2\pi t )\,\mathrm{Wb}\end{align}\]
So at \(t=0.5\,\mathrm{s}\) \(\mathcal{E}=0\,\mathrm{V}.\)
Magnetic Flux - Key takeaways
- Magnetic flux is defined as the amount of magnetic field flowing through a given surface area. It can be thought of as equal to the number of magnetic field lines passing through a surface area.
- For a fixed magnetic field the flux is defined as \(\Phi_B=\vec{B}\cdot\vec{A},\) where the area vector \(\vec{A}=A\vec{n}\) is directed perpendicular to the surface.
- For a varying magnetic field we use calculus to define the flux\[\Phi_B=\int_S\vec{B}\cdot\mathrm{d}\vec{A}.\]
- Faraday's Law states that the magnitude of an induced EMF in a closed loop of wire is proportional to the rate of change of magnetic flux through the surface enclosed by the wire.\[\mathcal{E}=-\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}.\]
References
- Fig. 1 - Flux line diagram, StudySmarter Originals.
- Fig. 2 - Iron ring and coils diagram, StudySmarter Originals.
- Fig. 3 - Turbine diagram, StudySmarter Originals.
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