Originally Posted by **oz93666** I was just wondering why this format pops up so often , always a half then one item , and another item squared ....
let me refresh my mind about **Dimensional analysis** .... everything can be looked at in terms of M (mass) ....L (length) ... and T ( time)
equations have to balance ...
energy measured in Joules has units M L2 T-2 (that's M ... L squared ... T to the minus 2)
velocity , measured in meters /sec has units L T-1 ...
so the formulae E= 1/2 m v2 analysed dimensionally has ...
M L2 T-2 on one side and M ... L T-1 .....L T-1 , on the other side ...both sides equal Inductance in farads has dimensions L !!! ... **LENGTH ** very curious Current has dimensions M1/2 L 3/2 T-2 !!!!! ( that's square root M ...L to the power 3 over 2 ... T to the minus 2) .... isn't that strange????
Energy in a inductor = 1/2 L I squared ....put the dimensions in and it all balances nicely ....!!! Capacitance in farads has dimensions L - 1 ( L to the minus 1) Voltage has dimensions M1/2 L 1/2 T-1
What does it all mean ???? |

I have highlighted the incorrect statements. Somehow or other you have lost the charge unit, Coulomb. (Or, if you like, the unit for current, Amperes.) Capacitance has the unit of farads in MKSA units, not inductance. Inductance is measured in henrys. It is little mentioned (at least I haven't heard it much) but capacitance is largely a geometric property of a collection of surface charges, so it should be somehow related to length units. (Yes, I'm not being very precise in saying that.) But capacitance only has units of length in the esu system of units.

Remember that Nature isn't perfect and Physicists have a tendency to define it to be perfect anyway. So similar formulas do tend to crop up fairly often. We also have things that come up from time to time without any great reason except that they turn out to be useful. Kinetic energy, for example, is defined in terms of mass and speed. The formula doesn't

*have* to be $\displaystyle 1/2 mv^2$ (it could be defined as $\displaystyle mv^4$ for example), but if we

*define* it as $\displaystyle 1/2 mv^2$ we get a nice little result: the work-energy theorem.

There are a number of properties that can be approximated by a

Taylor series (I don't know what your Math level is but it's first semester Calculus.) Say we define the property to be calculated as f(x). This can give a similar form as well when we are sitting at an equilibrium point, defined to be where the first derivative vanishes. The Taylor expansion then reads as

$\displaystyle f(x) \approx f(0) + \frac{1}{2}f''(0)x^2$

You can see the $\displaystyle AB^2$ form in the second term.

I don't know any other way to say it. The formulas are defined in such a way to be as useful as possible.

There is something else to mention. Many of these formulas are for perfect systems but as I said before, Nature isn't perfect. The energy stored in the magnetic field of an inductor is not always $\displaystyle 1/2LI^2$. This is only a "constitutive" definition of the magnetic field energy and can break down in certain materials. It's more of a first order approximation. Many of your formulas are defined that way.

-Dan