# Multiplying epsilon naught by a length quantity

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1. Jun 5, 2017

1. The problem statement, all variables and given/known data
Note that this formula ($C=4 \pi \epsilon_0 R$) and the others we have derived for capacitance involve the constant multiplied by a quantity that has the dimensions of a length.

2. Relevant equations
$\epsilon_0$ has the following units in SI:
$$\frac {C^2} {N \cdot m^2}$$ or $$\frac F m$$

3. The attempt at a solution
I don't know why the textbook states this? Does it mean it results in having the $F$ unit after canceling out the unit of length in SI? A friend told me that we also present $\epsilon_0$ in SI with $\frac H m$, thus multiplying by a length gives a result in $H$. I don't see any $\frac H m$ unit for epsilon naught on Wikipedia.

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2. Jun 5, 2017

### Staff: Mentor

A capacitance has to have F as unit.

The units can help to figure out formulas. Keeping the sphere as an example: The only relevant parameter is its radius, and $\epsilon_0$ of course. How can the capacitance depend on the radius? Well, we have F/m from $\epsilon_0$. The only way to get F is to multiply by the sphere radius: $\epsilon_0 R$. Up to a prefactor (here: $4 \pi$), we know already that this must be the capacitance of a sphere. Spheres have a capacitance proportional to their radius.

Henry is not a very compatible unit here. It is $1H = \frac{kg \,m^2}{C^2}$. Even if you consider 1/H, it still differs by m/s2 and there is no reasonable way to get rid of that difference.