- #1

- 304

- 14

- Homework Statement
- Attached below

- Relevant Equations
- Attached below

I don't know how to solve part c and d.

Attempt:

c) B_eff=B+e<M>

Substitute T_c into the equation in part b,

=> (B_eff-B)/e = Nμ_B tanh(B_eff/(N*e*μ_B))

Then?

Thank you.

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- Thread starter Clara Chung
- Start date

- #1

- 304

- 14

- Homework Statement
- Attached below

- Relevant Equations
- Attached below

I don't know how to solve part c and d.

Attempt:

c) B_eff=B+e<M>

Substitute T_c into the equation in part b,

=> (B_eff-B)/e = Nμ_B tanh(B_eff/(N*e*μ_B))

Then?

Thank you.

- #2

Mentor

- 8,237

- 5,117

Could you show what you got for part b?

- #3

- 304

- 14

Could you show what you got for part b?

I just rewrite the equation I found on part A to an equation containing B_eff only... is it what a self-consistency equation mean?...

- #4

Mentor

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A self-consistency equation for ##B_\textrm{eff}## means that you get an equation for ##B_\textrm{eff}## that depends on itself. You could solve it numerically by guessing the value of ##B_\textrm{eff}## in the hyperbolic tangent, then calculating ##B_\textrm{eff}## on the left-hand side, and iterate until the value of ##B_\textrm{eff}## doesn't change anymore.is it what a self-consistency equation mean?...

Your equation for (b) is correct. For (c), I think you can assume that ##B_\textrm{eff}## is small or, more properly

$$

\mu_\mathrm{B} B_\textrm{eff} \ll k T

$$

- #5

- 304

- 14

I did part c using your hintA self-consistency equation for ##B_\textrm{eff}## means that you get an equation for ##B_\textrm{eff}## that depends on itself. You could solve it numerically by guessing the value of ##B_\textrm{eff}## in the hyperbolic tangent, then calculating ##B_\textrm{eff}## on the left-hand side, and iterate until the value of ##B_\textrm{eff}## doesn't change anymore.

Your equation for (b) is correct. For (c), I think you can assume that ##B_\textrm{eff}## is small or, more properly

$$

\mu_\mathrm{B} B_\textrm{eff} \ll k T

$$

However in part d), how to deal with the cube term T^3/T_c^3?

- #6

Mentor

- 8,237

- 5,117

You didn't get the expected answer, which is given in the question. There is no need to keep the cubic terms.I did part c using your hint

- #7

- 304

- 14

Thank you, the answer I displayed above is for part d. The (T/T_C)^3 in the last line should be (T/T_C)^3/2. The expect term have no factor (T/T_C)^3/2, but there is one in my expression...You didn't get the expected answer, which is given in the question. There is no need to keep the cubic terms.

- #8

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$$

B_\textrm{eff} = e N \mu_\mathrm{B} \left( \frac{B_\textrm{eff} \mu_\mathrm{B}}{kT} - \frac{B_\textrm{eff}^3 \mu_\mathrm{B}^3}{3 k^3T^3} \right)

$$

which means you took

$$

\tanh(x) \approx x - \frac{1}{3} x^3

$$

Try instead ##\tanh(x) \approx x##.

- #9

- 304

- 14

However, in part d the question says expand it to 3rd order?

$$

B_\textrm{eff} = e N \mu_\mathrm{B} \left( \frac{B_\textrm{eff} \mu_\mathrm{B}}{kT} - \frac{B_\textrm{eff}^3 \mu_\mathrm{B}^3}{3 k^3T^3} \right)

$$

which means you took

$$

\tanh(x) \approx x - \frac{1}{3} x^3

$$

Try instead ##\tanh(x) \approx x##.

- #10

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Sorry, I was looking at part (c). The solution to (d) is not obvious to me, I'll have to think a bit.However, in part d the question says expand it to 3rd order?

- #11

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To get the equation given in the problem, I had to assume also that ##T/T_C \approx 1##, as stated in the problem.

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