Energy Formulas with the form (1/2)ab^2

1. May 13, 2012

krismath

1. The problem statement, all variables and given/known data
Can anybody help me think of Energy formulas/equations in the form below?
Any form of Energy.
(Please also state what does each variable stands for.)

2. Relevant equations

$E = \frac{1}{2}ab2 = \frac{1}{2}ac$

Where c = ab

3. The attempt at a solution

So far, this is what I came up with:
Kinetic Energy: $E = \frac{1}{2}mv2 = \frac{1}{2}pv$
M = Mass
V = Velocity
P = Momentum = M x V

Capacitor: $E = \frac{1}{2}CV2 = \frac{1}{2}QV$
C = Capacitance = Q/V
V = Voltage
Q = Charge

Spring : $E = \frac{1}{2}kx2 = \frac{1}{2}Fx$
K = Spring Constant
X = Extension of Spring
F = Force Acted = K x X

Self-inductance : $E = \frac{1}{2}LI2 = \frac{1}{2}∅_{B} I$
L = Self-Inductance
I = Current
B = Magnetic Flux = L x I

2. May 14, 2012

BruceW

Those all look like good examples to me. How many do you need to find?

3. May 14, 2012

krismath

It says: "As many as you can"

4. May 14, 2012

DeIdeal

How about the energy densities of electric and magnetic fields. They fit the "½ab²" requirement, but are not strictly speaking formulas of energy.

5. May 14, 2012

krismath

OK, I will try to look up that one (and post it in here, for reference to other people in the future that might need it.)

6. May 14, 2012

flatmaster

The energy density term in Bernoulli's equation would also fit into this category.

7. May 14, 2012

krismath

Okay, now I found it,

Electric Field Energy Density: $Energy (per volume) = \frac{1}{2}\ \epsilon_{o}\ E^{2}$
$\epsilon_{o}\ =\ 8.854187817\ \times\ 10^{-12}\ F\ m^{-1}$ = Electric Constant
E = Electric Field

Magnetic Field Energy Density: $Energy (per volume) = \frac{1}{2}\ \frac{1}{ \mu_{o}\ }\ B^{2}$

$\mu_{o}\ =\ 4\pi\ \times\ 10^{-7}$ = Magnetic Constant
B = Magnetic Field

Last edited: May 14, 2012
8. May 14, 2012

krismath

Hmmm... But on what I have learned in my class, that term is derived from the Kinetic Energy, isn't it?

9. May 14, 2012

DeIdeal

Yeah, those were the ones I was talking about. And if you want to write them like this:

$E = \frac{1}{2}ab^{2} = \frac{1}{2}bc,\quad c=ab$

You can do so:

$u_{E} = \frac{1}{2}\epsilon_{0} E^{2} = \frac{1}{2} \vec E \cdot \vec D ,\quad D =\epsilon_{0} E$

$u_{B} = \frac{1}{2}\frac{1}{\mu_{0}} B^{2} = \frac{1}{2} \vec B \cdot \vec H ,\quad H =\frac{B}{\mu_{0}}$

Where D and H are the electric displacement field and the strength of a magnetic field, respectively, if you haven't seen them before.

10. May 14, 2012

krismath

Thank you very much!