## The Astrophysics of Faraday’s Law of Induction

Posted in astrophysics, physics with tags , , , on February 28, 2010 by Grad Student

Something I love about physics is how the simple intuition you gain from an introductory physics level discussion of lenz’s law and Faradays law of induction can enable you to understand one of the outstanding problems in theoretical astrophysics: how are large scale celestial magnetic fields generated?

In terms of magnetic fields, Faraday’s law of induction says that any change in the magnetic field through a conducting loop will induce currents in that loop that will oppose such a change in the magnetic field.  In other words, changing the magnetic field in an area where currents can flow is hard and therefore takes some time.

Alternatively, Faraday’s law says that changing the current in a circuit is hard and takes time.  This is because the current itself generates a magnetic field.  So, changing the current will change the magnetic field through the loop.  So Faraday tells us that there will be an induced current that opposes the original attempt to increase (or decrease) the current.  Upshot:  changing the current in a loop is hard and therefore takes time.

Approximately how long does it take for the magnetic field to go from zero to the equilibrium value, B?  For a simple circuit with resistance and some inductance (an RL ciruit) any student of introductory physics could answer this question.

$t \sim \frac{L}{R}$,

where L is a measure of inductance of the circuit and R is the circuit’s resistance.  The inductance of a circuit is simply a measure of how much the circuit resists any change in the magnetic field, or by another measure, it is the measure of how difficult it is to change the amount of current flowing in the circuit.

In astrophysics, there are similarities to a circuit with resistance and inductance*.

The Sun’s magnetic field is about the strength of Earth’s magnetic field, 0.0001 Tesla (where Tesla is the SI unit of magnetic field strength).  Further, the Sun is composed of ionized gas and therefore it is very conductive, meaning electric currents can flow easily.  Thus we can ask the same question about the Sun that we asked about the circuit: how long does it take for the magnetic field to attain an equilibrium value of 0.0001 Tesla?  As it turns out L/R is a good way of estimating this timescale, all we need to do is figure out how to estimate the inductance and the resistance of the Sun.

The inductance, L, depends on an important quantity known as the magnetic flux.  The magnetic flux can be thought of as the number of magnetic field lines that pierce through a surface.  In the case of the Sun, the flux is approximately

$\Phi\sim B D^2$,

where the greek letter Phi is the flux, B is the sun’s mean magnetic field, and D is the diameter of the sun.  Therefore, the number of magnetic field lines threading the Sun is the average magnetic field multiplied by the cross sectional area of the sun.  Now we can define the inductance as

$L=\frac{\Phi}{I}$,

where I is the total current that is maintaining the magnetic field of the Sun.  If we recognize that the magnetic field in the Sun is approximately

$B \sim\frac{\mu_0I}{D}$,

then we can calculate the inductance by using the expression we have for the magnetic flux, Phi, and magnetic field, B:

$L=\frac{B D^2}{I}=\mu_0 D$.

Estimating the resistance is also possible using introductory physics:

$R\sim \rho \frac{D}{A}\sim \frac{\rho}{D}$,

where the greek letter rho, the resistivity, is a measure of how much the Sun’s plasma prevents currents from flowing by particle-particle collisions.  Unfortunately, we can’t use introductory physics to calculate the resistivity, so I’ll just note that it’s proportional to the temperature of the plasma to the -3/2 power**.  Plugging everything in to the original formula for T, we finally get:

$t=\frac{\mu_0 D^2}{\rho}\sim\frac{D^2}{4 \times 10^{16}}T^{3/2}$ years.

Plugging in the diameter and average temperature of the Sun we get:

$t\sim\frac{(1.4 \times 10^9 m)^2}{4 \times 10^{16}}(1,000,000 K)^{3/2}\sim50$ billion years.

Clearly we’re missing something here, the universe is only 14 billion years old.  If the Sun was born with very little magnetic field 5 billion years ago, how did it build up a field of 0.0001 Tesla against induction in that time?  According to Faraday’s law and a little plasma physics it takes much longer for such a field to reach its equilibrium value.  Another huge problem is that the Sun’s magnetic field reverses every 12 years or so.  That means the real time scale for significant changes in the Sun’s magnetic field is a decade, not 5 billion years.

The answer to this conundrum lies in a topic that is still very much at the forefront of astrophysics research today: dynamo theory.  Dynamo theory is an area of plasma physics that arose because the simple arguments I’ve presented here do not match with observations of the magnetic fields originating from the Sun, Earth, or even the galactic disk.

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References:

* Check out the pgs 4-5 in Plasma Physics for Astrophysics by Kulsrud (2005) for a similar explanation of the magnetic induction timescale in astrophysics (these pages can be viewed for free using amazon’s “look inside” feature).  The most common explanation of the induction timescale (not presented here) comes from the resistive induction equation (on pg 112 of the previous link) when the fluid velocity is zero.

** The -3/2 power of the temperature dependence for the plasma resistivity can seen if it’s recognized that the resistivity is proportional to the plasma collision frequency.  It’s also a bit surprising that the resistivity doesn’t depend on the density of particles.  The answer is that the more particles the more current, but the more particles the more particle-particle collisions which kill the current.  These two effects approximately cancel each other.  A more detailed analysis reveals that the resistivity weakly depends on the density (logarithmically) .