How would you derive n(E)= E^2/((e^E/T) - 1)
a) Evaluate dn/dE at fixed T. Leave answer in symbolic forms of E and T.
b) dn/dT at fixed E.
To derive the expression for n(E) = E^2/((e^(E/T)) - 1), we need to find the derivative with respect to E and T separately. Let's start with part (a).
a) To evaluate dn/dE at fixed T, we need to find the partial derivative of n(E) with respect to E while keeping T constant.
Step 1: Write down the expression for n(E): n(E) = E^2/((e^(E/T)) - 1)
Step 2: Take the derivative of n(E) with respect to E while treating T as a constant. We can use the chain rule here.
Chain rule:
d/dE(e^(f(E))) = e^(f(E)) * df/dE
For our case, f(E) = E/T.
Using the chain rule, we get:
dn/dE = (2E/T) * (e^(E/T))/((e^(E/T)) - 1)^2
So, the derivative dn/dE at fixed T is given by:
dn/dE = (2E/T) * (e^(E/T))/((e^(E/T)) - 1)^2
b) To evaluate dn/dT at fixed E, we need to find the partial derivative of n(E) with respect to T while keeping E constant.
Step 1: Write down the expression for n(E): n(E) = E^2/((e^(E/T)) - 1)
Step 2: Take the derivative of n(E) with respect to T while treating E as a constant. Again, we can use the chain rule.
Chain rule:
d/dT(e^(f(T))) = e^(f(T)) * df/dT
For our case, f(T) = E/T.
Using the chain rule, we get:
dn/dT = (-E^2/T^2) * (e^(E/T))/((e^(E/T)) - 1)^2
So, the derivative dn/dT at fixed E is given by:
dn/dT = (-E^2/T^2) * (e^(E/T))/((e^(E/T)) - 1)^2
These expressions can be used to evaluate the derivatives of n(E) with respect to E and T at any fixed values of E and T, respectively.