What Is Energy? — Question 7

Question 7

An internal transfer disappears from the combined ledger

For this introductory problem use the supplied accounting rule $\Delta E=E_{\mathrm{in}}-E_{\mathrm{out}}$, with all quantities in joules. Two connected modules A and B initially contain 100 J and 40 J. During an interval A receives 30 J from outside, sends 20 J to B, and B releases 12 J outside. No other transfers occur.

1. Find the final energy of each module.

2. Find the final energy of A+B using only its outer boundary.

3. Explain the error in counting the 20 J as an additional input to A+B.

Question 7 - Solution

An internal transfer disappears from the combined ledger

(a) Write separate signed balances.

The governing relation is $$\boxed{\Delta E_A=I_A-T_{AB},\qquad\Delta E_B=T_{AB}-O_B}$$

$$E_{A,f}=100+30-20=110\ \mathrm J,\qquad E_{B,f}=40+20-12=48\ \mathrm J.$$

(b) Cancel the shared transfer.

The pair begins with 140 J and gains $$30-12=18\ \mathrm J,$$ ending at 158 J. This equals $110+48\ \mathrm J$.

(c) Identify the double count.

The 20 J leaving A is exactly the 20 J entering B. It crosses a boundary internal to A+B and supplies no new energy to the pair. Counting it as external would falsely predict 178 J.