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

  1. May 13, 2012 #1
    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


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

    Where c = ab

    3. 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
     
  2. jcsd
  3. May 14, 2012 #2

    BruceW

    User Avatar
    Homework Helper

    Those all look like good examples to me. How many do you need to find?
     
  4. May 14, 2012 #3
    It says: "As many as you can"
     
  5. May 14, 2012 #4
    How about the energy densities of electric and magnetic fields. They fit the "½ab²" requirement, but are not strictly speaking formulas of energy.
     
  6. May 14, 2012 #5
    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.)
     
  7. May 14, 2012 #6
    The energy density term in Bernoulli's equation would also fit into this category.
     
  8. May 14, 2012 #7
    Okay, now I found it,

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

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

    [itex]\mu_{o}\ =\ 4\pi\ \times\ 10^{-7}[/itex] = Magnetic Constant
    B = Magnetic Field
     
    Last edited: May 14, 2012
  9. May 14, 2012 #8
    Hmmm... But on what I have learned in my class, that term is derived from the Kinetic Energy, isn't it?
     
  10. May 14, 2012 #9
    Yeah, those were the ones I was talking about. And if you want to write them like this:

    [itex] E = \frac{1}{2}ab^{2} = \frac{1}{2}bc,\quad c=ab [/itex]

    You can do so:

    [itex] u_{E} = \frac{1}{2}\epsilon_{0} E^{2} = \frac{1}{2} \vec E \cdot \vec D ,\quad D =\epsilon_{0} E [/itex]

    [itex] 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}} [/itex]

    Where D and H are the electric displacement field and the strength of a magnetic field, respectively, if you haven't seen them before.
     
  11. May 14, 2012 #10
    Thank you very much!
     
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