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

[krismath]

Homework Statement

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.)

Homework Equations

[itex]E = \frac{1}{2}ab² = \frac{1}{2}ac[/itex]

Where c = ab

The Attempt at a Solution

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

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

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

Self-inductance : [itex]E = \frac{1}{2}LI² = \frac{1}{2}∅_{B} I[/itex]
L = Self-Inductance
I = Current
∅_B = Magnetic Flux = L x I

 

[BruceW]

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

 

[krismath]

It says: "As many as you can"

 

[DeIdeal]

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

 

[krismath]

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

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.)

 

[flatmaster]

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

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

 

[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

 

[krismath]

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

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

 

[DeIdeal]

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

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.

 

[krismath]

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.

Thank you very much!