Confusion between magnetic field and magnetic flux

Question

I've been learning about electromagnetism and Maxwell's equations (in integral form), and I'm slightly confused.

The Ampere-Maxwell law (as I know it):

$$ \oint\vec{B}\cdot\vec{s}=\mu_0I+\mu_0\epsilon_o\frac{d\Phi_e}{dt} $$

(Integral of magnetic field around loop is $\mu_0$ times the current through the loop plus $\mu_0\epsilon_0$ times the change in electric flux through the loop)

So when we have the Biot-Savart Law, it tells us that there is a vector in every point in space that describes the magnetic field. However, when using the Maxwell-Ampere Law, what is it telling? What is the $\vec{B}$ in the equation, and how do you use it? Is it telling the sum of the magnitude field vectors around the loop (i.e. every point on the loop)? If so, can I use it to find the magnetic field at a certain point somehow?

Answer 1

I understand your confusion here, and I had the same concern when I learned it at the first time. Since the left-hand side is an integral of B, at best we can obtain the flux through some loop, not the value of B at some certain point. Is this your question?

Note that the equation do contain some more information. It holds for ANY loop. Intuitively speaking, this "any" allows you to draw a loop at small as possible in which the magnetic field does not vary much. In this way, when the flux through this loop is obtained, one can obtain the magnetic field simply by dividing the flux by the area.

This is the intuitive understanding of Stokes' Theorem (including Green's Theorem, Gauss Theorem and Stokes' Theorem in the usual sense). Using that, the integration equation you wrote down can be transformed into a differential equation (sorry, this is my first time answering questions here and I haven't figure out how to type down equations...). The Biot-Savart equation is just the solution of this differential equation in some particular configuration (steady current, no time-varying electrial field and so on.)

In conclusion, the equation you wrote down applies to any loop and this arbitraryness allows you to deduce B out of flux (or transform the integration equation to a differential equation). Bio-Savart Law is just a particular solution to that differential equation subject to some boundary condition.

Hope it helps.

Answer 2

To add to Lordrain's answer, I'll give you two scenarios where this equation would be useful. First, notice that this is Maxwell's extension of Ampere's law, meaning you can apply it to situations where a magnetic field exists but there is no current (e.g inside a capacitor with an alternating current). Let's first consider the situation where you'd like to find the magnetic field produced by a current carrying wire. We will take the Amperian loop to be a concentric circle around the wire. We know that:

$$\oint\vec{B}\cdot d\vec{s}= 2\pi rB$$

since the magnetic field is always parallel to the loop (by the right hand rule), and the path length is $2\pi r$ where $r$ is the distance from the wire. We also know that:

$$\mu_0I+\mu_0\epsilon_o\frac{d\Phi_e}{dt} = \mu_0I$$

since there is no displacement current in this situation. This allows us to find an equation for the magnitude of $B$:

$$B=\frac{\mu_0I}{2\pi r}$$

Now let's consider the case of finding the magnetic field inside a capacitor $(r<R)$ with two circular plates or radius $R$, and with an alternating current in the circuit. Taking the same Amperian loop as before, we also have:

$$\oint\vec{B}\cdot d\vec{s}= 2\pi rB$$

However, in this situation, we have:

$$\mu_0I+\mu_0\epsilon_o\frac{d\Phi_e}{dt} = \mu_0\epsilon_o\frac{d\Phi_e}{dt}$$

because we know that there is no current through a capacitor.

If we take $E=E_0 sin(\omega t)$ for an alternating current, we know that $\Phi_E =AE_0 sin(\omega t)=\pi r^2 E_0 cos(\omega t) $ where $A$ is the area of the loop, and $r$ is its radius. Thus, $\frac{d\Phi}{dt}=\omega \pi r^2 E_o cos(\omega t)$, giving:

$$2\pi rB = \mu_0\epsilon_o\frac{d\Phi_e}{dt}= \mu_0\epsilon_o \omega \pi r^2 E_o cos(\omega t)$$

and we obtain the final expression for $B$:

$$B=\frac{1}{2} \mu_0 \epsilon_0 \omega r E_0 cos(\omega t)$$

Notice that in these two cases, the line integral wasn't evaluated mathematically, but rather arguments of symmetry (and a good choice of Amperian loop) were used to reduce it to something easier to work with. Hopefully this gives you an idea of how the Ampere-Maxwell law can be used to find the magnitude of the magnetic field at certain places in different situations.