# Question on units for voltage equation of a voltage at a distance z from a single line charge

I have a very basic question on the units for the equation of the potential at a distance: $$z$$ from a single line charge.

The electric potential due to a very long line charge at some distance $$z$$ is given by:

$$V(z)=-\frac{\lambda}{2\pi\epsilon_0}\ln(z)+C$$

If I look at the units of the terms, $$\lambda$$ is a line charge; so the units are $$\frac{C}{m}$$.

The denominator: $$2\pi\epsilon_0$$ is expressed in units of $$\frac{F}{m}$$.

This makes the units of $$\frac{\lambda}{2\pi\epsilon_0}$$: $$\frac{C}{F}$$ = $$V$$.

This implies that the term $$\ln(z)$$ is unitless and that our constant $$C$$ is in $$V$$.

For this to be true; $$z$$ must be a ration of distances; so that the meters cancel.

Would a verbose version of this equation be:

$$V(z)=-\frac{\lambda}{2\pi\epsilon_0}\ln(\frac{z}{1})+C$$

• The second log is hidden in the constant C, which will be proportional to $\log{(z_0)}$ which is your reference point of some known potential, the log will combine and be the logs of a ratio of distances. – Triatticus Nov 9 '18 at 16:45

Yes, and in fact this isn't just true for this specific case, but it should be true in general that any argument of $$\ln$$ should be dimensionless.
One intuitive reason for this was explained to me by my professor, when he said to consider what might happen if you Taylor expand $$\ln(1+x)$$ (using this for simplicity):
$$\ln(1+x)=x-\frac{1}{2}x^2+\frac{1}{3}x^3-\frac{1}{4}x^4+...$$
If $$x$$ is not dimensionless, then (a) you shouldn't be adding it to $$1$$ in the first place, but more importantly (b) the dimensions of the series expansion explode to $$\infty$$, which is not good!
Many people (myself included sometimes) are guilty of writing things like $$\ln(z)$$, and I think it's fine in most cases, as long as you're aware that you should actually have a reference length (e.g. of $$z=1\text{ m}$$) implicitly included in the argument.
Note that this is also true for arguments of $$\sin$$, $$\cos$$, $$\exp$$ etc - the arguments should always be dimensionless.