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Energy stored in velocity and capacitors

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cduma

Tue Dec 01 2009, 02:28AM

Registered Member #1822 Joined: Fri Nov 21 2008, 08:04PM
Location:
Posts: 300

Is it a coincidence that the formula for determining the energy in joules stored in a capacitor is very similar to determining the energy in a moving object? Is there a reason why they are closely related?
Capacitor: V^2*.5(F)=J
Velocity: MpS^2*.5(Kg)=J

Turkey9

Tue Dec 01 2009, 03:04AM

Registered Member #1451 Joined: Wed Apr 23 2008, 03:48AM
Location: Boulder, Co
Posts: 661

I'm not sure on the exact science of it but there are a lot of coincidences like that in physics land. I think it might have something to do with how the units were allocated and how the different variables were defined to make it all much easier. Just like one cm^3 is one mL and one mL of water weighs one gram.

In a capacitor, F describes the amount of charge, and in kinetics Kg describes the amount of matter. The moving charge is what makes the energy in the electrical problem and likewise it's the moving mass that makes the energy in a kinetics problem. Voltage can also be thought of as the velocity of electricity. The higher the potential, the more energy. Same as with velocity. Maybe someone else can put my ramblings into more concise words

klugesmith

Tue Dec 01 2009, 04:04AM

Registered Member #2099 Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1716

Good analogy.
We get square functions because each uniform increment of voltage [velocity]
adds energy that is proportional to the voltage [velocity].

One way to derive the formulas is based on energy (work) = the integral of power with respect to time.
The power (at constant charging current or accelerating force) goes up linearly with time.

Now with formulas:
W work
P power
C capacitance
i current
V voltage
m mass
f force
v velocity

Electrical:
W = integral( P dt) = integral( i(t) V(t) dt ).
with constant i, V(t) = i * t / C.
So W = i^2/C integral( t dt ) = i^2/C * t^2/2 = C V^2 / 2.

Physical:
W = integral( P dt) = integral( f(t) v(t) dt ).
with constant f, v(t) = f * t / m.
So W = f^2/m integral( t dt ) = f^2/m * t^2/2 = m v^2 / 2.

Q.E.D.

Bored Chemist

Tue Dec 01 2009, 06:58AM

Registered Member #193 Joined: Fri Feb 17 2006, 07:04AM
Location: sheffield
Posts: 1022

Similar equations are found for the energy in a current carrying inductor or a stretched spring.

Dr. Slack

Tue Dec 01 2009, 08:26AM

Registered Member #72 Joined: Thu Feb 09 2006, 08:29AM
Location: UK St. Albans
Posts: 1659

Good observation, I agree with all of the other posters, just want to put my own slant on the explanation.

Energy is always the product of a "force-like" thing and a "distance-like" thing. Obviously in the case of electrical things, units of V and A etc have to be incorporated, but their product still ends up as having dimensions of energy.

When the force and distance are independant, like height and gravitational attraction, then we get the energy = mgh type formula.

However, when force and distance are related by a "spring constant" type proportionality, then we can can still use the product of both terms, or use one of them squared with the spring constant in there. Because we are integrating a linear variable to find the total energy, we always end up with a square term, and usually also have a constant of 1/2 in there because of the way the integral works.

Examples are

speed and momentum for kinetic energy (related by mass)
voltage and charge (related by capacitance)
current and flux (related by inductance)
B field and H field (related by u0)
pressure and change in volume (related by mass, gas constant, temperature)
torque and angular speed (related by moment of inertia)
spring force and deflection (related by spring constant)

Once you have a formula for the energy in a system, you are 90% of the way to understanding it.

klugesmith

Tue Dec 01 2009, 03:21PM

Registered Member #2099 Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1716

Bored Chemist wrote ...

similare equations are found for the energy in a current carrying inductor or a stretched spring.

BC, you beat me to it. For easiest digestion, try associating capacitor with spring (potential energy). and inductor with moving mass (kinetic energy).

When we connect the capacitor and inductor, or mass and spring, we get analogous second-order formulas for harmonic oscillation -- resonance -- which is also ubiquitous in Nature.

ShawnLG

Tue Dec 01 2009, 06:12PM

Registered Member #286 Joined: Mon Mar 06 2006, 04:52AM
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Posts: 399

I think all this is related to Einstein's Massâ€“energy equivalence formula. E = mc2

Steve Conner

Tue Dec 01 2009, 06:28PM

Registered Member #30 Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706

A cookie (and possibly even a Nobel Prize ;) ) for anyone who can explain why Einstein's equation doesn't have the 1/2 in it.

Turkey9

Tue Dec 01 2009, 11:32PM

Registered Member #1451 Joined: Wed Apr 23 2008, 03:48AM
Location: Boulder, Co
Posts: 661

E=mc^2 doesn't have a velocity component, does it? It's for determining the energy released in nuclear reactions when some mass is turned directly into energy. c^2 is a huge constant (9x10^16) so that's why so much energy can be released when only a tiny amount of mass is used. Einstein has a different equation for dealing with velocity from zero to approaching the speed of light, right?

mikeselectricstuff

Wed Dec 02 2009, 12:36AM

Registered Member #311 Joined: Sun Mar 12 2006, 08:28PM
Location:
Posts: 253

Steve McConner wrote ...

A cookie (and possibly even a Nobel Prize ;) ) for anyone who can explain why Einstein's equation doesn't have the 1/2 in it.

When there's that much energy flying about, who's botherd about the odd 50% if it makes the formula look more snappy..:-p

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