# Total Flux Around a Long Straight Conductor

The magnetic flux density around a long straight conductor is given, B = $$\dfrac{\mu_0\cdot I}{2\pi R}$$

I have integrated this expression (along a conductor of length l) in order to calculate the flux contained within distance R, $$\phi = \dfrac{\mu_0 \cdot I \cdot l}{2\pi }( ln(R)- ln(0))$$

However $$ln(0)$$ is undefined and therefore this expression cannot be evaluated. This comes from the flux density being infinite when $$R = 0$$. Have I made a mistake here?

Additionally, is it possible to use this formula to calculate the total flux (i.e. integrating from $$R = 0$$ to $$R = infinity$$ ). This formula would imply that the total flux around the conductor is infinite.

Is there an alternative way to calculate the total flux around a long straight conductor? Perhaps something along the lines of $$\phi = I \cdot L$$ (where I is the current in the conductor and L its inductance)

• I've added the homework-and-exercises tag. In the future, please use that tag on this type of question. – Ben Crowell Oct 2 '18 at 15:27
• Why are you integrating with respect to $R$? – Aaron Stevens Oct 2 '18 at 15:27
• I'm integrating with respect to R as I'm trying to calculate the flux by summing the flux density along distance R – Chris Brown Oct 2 '18 at 15:33
• I'm trying to create a model for something I have built in the lab so I didn't use the homework-and-exercises tag @BenCrowell – Chris Brown Oct 2 '18 at 15:34
• What you have forgotten is that inside the wire the magnetic field drops linearly to zero at the centre. – Farcher Oct 2 '18 at 16:06

I assume that you want to calculate flux $$\Phi=\int BdS$$ across the rectangle with dimensions $$R$$ and constant $$l$$ which is outside of the conductor. Then $$\Phi=\int_{R_1}^{R_2}\frac{\mu_0 I}{2\pi R}ldR$$.
I don't think there is any problem with infinity. If $$R$$ grows, the $$\Phi$$ should grow too, although very slowly (which can be seen by looking at the logarithm at your result.
P.S. If it's that $$1/R$$ in $$B$$ bothering you, try looking at this: $$\sum\limits_{i=1}^\infty \frac{1}{n} = \infty$$ Sometimes even sum of infinite series with decreasing numbers can be infinite.
• For conductor of length $l$, I want to know the total flux around it. Therefore the area is $l *2*\pi*R$ (surface of a cylinder) and this should be integrated for $R=0$ to $R=infinity$ – Chris Brown Oct 3 '18 at 13:05
• Well then the total flux is zero. $\vec B$ will look like in this picture google.com/…: and the cylinder you are talking about is going to be parallel to $\vec B$ at every point of its surface. Normal vector of the cylinder is going to be perpendicular to $B$ everywhere and therefore $\Phi=\int \vec{B} d \vec{S} = 0$ – Andrej Oct 3 '18 at 13:34