I don't understand this equation

[member 392791]

So my professor has been using an equation to find the magnetic field of a wire, but the equation is not present in the textbook. it is

B = μI/(4πa)(cosθ_1 - cosθ_2)

The theta is is an interior angle and theta 2 is an exterior angle. Anyone know what I'm talking about?

 

[mathsciguy]

You might want to give more details, what is the orientation of the wire? This might be a circular wire, but we are not sure. Where are those angles measured from?

 

[kontejnjer]

I believe this equation gives the magnitude of the magnetic field from a straight wire of finite length where $\theta_1$ is the interior angle from one end point, $\theta_2$ is the exterior angle from the other end point, and $a$ is the perpendicular distance from the wire. Also, notice that when $\theta_1 \rightarrow 0$ and $\theta_2 \rightarrow \pi$, you obtain the field due to an infinite straight wire.

 

[member 392791]

What kontejnjer is describing is the right equation. But, where does this thing come from?

 

[raopeng]

I did a rough calculation and the results seem to be right. The idea is to integrate the contribution of one infinitesimal of wire to the magnetic field at the point. So we write $\int{dB}=\int{\frac{Idl sinw}{r^2}}$(I omit the constants) as from the Biot-Savart law(thanks to Philip Wood). Trying to express dl as a function of dw and integrating the whole thing should give the desired result.

 

[Philip Wood]

The equation raopeng quotes is the Biot-Savart rule (or law). This can be deduced from two of Maxwell's equations when $\frac{\partial \textbf{E}}{\partial t} = 0$. The integration on the right hand side is around the whole circuit.

Search for Biot-Savart on the internet. Several sites give introductions.

 

[Meir Achuz]

That is Eq. (7.13) of "Classical Eletromagnetism" by Franklin, which also gives its derivation.
The key is that the usual derivation for an infinite length wire involves an integral from -infinity to +infinity.
Using finite limits gi.oves your equation.

 

[DrZoidberg]

What's the interior and exterior angle of a straight wire?
Also if you take the equation B = μI/(4πa)(cosθ_1 - cosθ_2) and set cosθ_1=0 and cosθ_2=π you get B = -μI/(4a) which is not the field of an infinite wire. Could someone clear this up please?

 

[raopeng]

I believe it is θ_1=0 and θ_2=π, in that case cosθ_2=-1 and cosθ_1=1, so the result will be μI/(2πa). The plus minus sign corresponds to the direction of the magnetic field(in or out of the plain of the wire).