# Multiplying epsilon naught by a length quantity

## Homework Statement

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.

## Homework Equations

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

## 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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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.

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