A source of emf \(\&\) has internal resistance \(r\), (a) What is the terminal voltage when the source supplies a current \(I ?\) (b) The net power supplied is the terminal voltage times the current. Starting with \(P=I \Delta V,\) derive Eq. \((18-22)\) for the net power supplied by the source. Interpret each of the two terms. (c) Suppose that a battery of emf 8 and internal resistance \(r\) is being recharged: another emf sends a current \(I\) through the battery in the reverse direction (from positive terminal to negative). At what rate is electric energy converted to chemical energy in the recharging battery? (d) What is the power supplied by the recharging circuit to the battery?

Step by step solution

01

(a) Terminal Voltage Calculation

To calculate the terminal voltage, we need to consider the emf and the internal resistance of the source. The relationship between them is given by: $$V_{terminal} = \& - Ir$$ Here, \(\&\) is the emf, \(I\) is the current, and \(r\) is the internal resistance. Now, we can plug in the given values and find the terminal voltage.

02

(b) Deriving Net Power Supplied Equation

To derive the equation for the net power supplied, we need to start with the given formula: $$P = I V_{terminal}$$ Now, we can substitute the expression for terminal voltage from part (a): $$P = I (\& - Ir)$$ Expanding this expression, we get: $$P = I\& - I^2r$$ This is the required equation for the net power supplied by the source. The first term \(I\&\) represents the total power produced by the source, and the second term \(I^2r\) represents the power lost as heat due to the internal resistance.

03

(c) Rate of Electric Energy Conversion to Chemical Energy

In the case of battery recharging, the current is sent in the reverse direction. We can denote this by \(-I\), and the terminal voltage by \(-V_{terminal}\). The terminal voltage will still be given by: $$-V_{terminal} = \& + Ir$$ The rate of electric energy conversion to chemical energy is the same as the power supplied by the recharging circuit, which is the absolute value of the product of current and terminal voltage: $$P = |-I|-|V_{terminal}|$$ Substituting the expression for \(-V_{terminal}\), we get: $$P = |-I|(\& + Ir)$$ This gives us the rate at which electric energy is converted to chemical energy during recharging.

04

(d) Power Supplied by the Recharging Circuit

The power supplied by the recharging circuit to the battery is given by: $$P_{supplied} = |-I| (\& + Ir)$$ This expression is the same as the one we found in part (c) because the power supplied by the recharging circuit goes into converting electric energy to chemical energy in the battery. So, we have: $$P_{supplied} = |-I|(\& + Ir)$$

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