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Math Help - Find an expression for electric field

  1. #1
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    Find an expression for electric field

    Hi ,please I need something to finish this problem:

    The specific heat s of a material in [J/(kg deg C] is the amount of energy in joules required to raise the temperature of 1[kg] of material by one degree C. The density ρ of a material in [kg/m3] is the mass in [kg] per cubic meter. If a current density J exists inside a material for a time Δt, show that the rise in temperature Δ T in degree C given the formula (σ is conductivity of the material):

    Δ T=(J^2* Δ t)/(s* σ* ρ)


    This is how I started:

    we know
    J=σ*E (E electric field) --> E=J/σ

    Now the power density.
    Pd=J*E
    Pd=J*J/σ=J^2/σ

    to have the enegy density(Eg), I multiply by Δ t (t is the time).
    Eg=Pg*Δ t=J^2/σ

    Then I said that: ΔT=Eg/s
    where s is the specific heat and T the temperature.

    --> ΔT=J^2*Δt/(s* σ)

    Now since I have been woking with the energy density. To have the energy, I divide by the density ρ.
    If I do that I got the expression, but i would like to know if my reasonning is right.
    B.
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  2. #2
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    Quote Originally Posted by braddy View Post
    Hi ,please I need something to finish this problem:

    The specific heat s of a material in [J/(kg deg C] is the amount of energy in joules required to raise the temperature of 1[kg] of material by one degree C. The density ρ of a material in [kg/m3] is the mass in [kg] per cubic meter. If a current density J exists inside a material for a time Δt, show that the rise in temperature Δ T in degree C given the formula (σ is conductivity of the material):

    Δ T=(J^2* Δ t)/(s* σ* ρ)


    This is how I started:

    we know
    J=σ*E (E electric field) --> E=J/σ

    Now the power density.
    Pd=J*E
    Pd=J*J/σ=J^2/σ

    to have the enegy density(Eg), I multiply by Δ t (t is the time).
    Eg=Pg*Δ t=J^2/σ

    Then I said that: ΔT=Eg/s
    where s is the specific heat and T the temperature.

    --> ΔT=J^2*Δt/(s* σ)

    Now since I have been woking with the energy density. To have the energy, I divide by the density ρ.
    If I do that I got the expression, but i would like to know if my reasonning is right.
    B.
    Your line for Eg should read:
    Eg=Pg*Δ t=J^2*Δ t/σ
    I assume it's a typo since you have the correct expression on the next line.

    I have two problems with this derivation. I can't finish your problem because I don't know the answer to the second.

    My first objection is your notation:
    Eg/ρ is the energy density<-- Yes.

    But you said
    ΔT = Eg/s
    which is not true.
    ΔT = [(Eg/ρ)*ρ]/s = [total energy input]/[specific heat]
    (Unless "s" is the specific heat density??)

    The second objection is that you are assuming that ALL the energy due to the current flow is absorbed by the material. I find this assumption to be realistically flawed unless ALL of the current's energy is absorbed by the material: ie. we are speaking of a perfect insulator. (In which case σ = 0, which is obviously not going to be true since that would cause ΔT to go to infinity.)

    However I will note that I suspect your derivation is intended to be the correct one because the equation you are trying to derive ( Δ T=(J^2* Δ t)/(s* σ* ρ) ) makes no mention of the material's thermal reaction to a current, as it would need to if my objection were correct.

    -Dan
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