### Physics Problem: Electrical Power in a Circuit**Problem Statement:**An external resistor with resistance \( R \) is connected to a battery that has electromotive force (emf) \( \mathcal{E} \) and internal resistance \( r \). Let \( P \) be the electrical power output of the battery. By conservation of energy, \( P \) is equal to the power delivered to the external resistor.**Diagram Explanation:**The diagram shows a circuit containing a battery with internal resistance \( r \) and emf \( \mathcal{E} \), connected in series with an external resistor \( R \). This setup is used to analyze how power is distributed in the circuit.**Questions:**(a) Find \( P \) in terms of \( \mathcal{E}, r, \) and \( R \).(b) If \( \mathcal{E} = 40\, \text{V} \) and \( r = 2\, \Omega \), calculate the power delivered to the external resistor for six different values of \( R \): 0.25 \( \Omega \), 1 \( \Omega \), 1.75 \( \Omega \), 2.5 \( \Omega \), 3.25 \( \Omega \), and 4 \( \Omega \).(c) In Part (b), you should have found that \( P \) does not increase monotonically as \( R \) increases. Using the equation for \( P \) from Part (a), find the value of \( R \) in terms of \( r \) such that \( P \) is maximized. What is the maximum value of \( P \) in terms of \( \mathcal{E} \) and \( r \)?---**Solution Guide:**1. **Finding \( P \):** - Use equations relating voltage, current, and resistance to express \( P \) in terms of the given variables. 2. **Calculating Power for Given \( R \) Values:** - Substitute the values of \( \mathcal{E}, r, \) and various \( R \) into the formula to find the power delivered in each case. 3. **Maximizing \( P \):** - Use calculus or algebraic manipulation to derive the conditions under which \( P \) is maximized. - Typically, this involves setting the derivative

We are given a battery with some internal resistance. Both these internal resistance and external…

Answered: 2. An external resistor with resistance…

Similar questions

(a) What is the terminal voltage (in V) of a large 1.90 V carbon-zinc dry cell used in a physics lab to supply 1.20 A to a circuit, if the cell's internal resistance is 0.200 Q? V (b) How much electrical power (in W) does the cell produce? W (c) What power (in W) goes to its load? W
A capacitor with a capacitance of 3.5 μF is initially uncharged. It is connected in series with a switch of negligible resistance, a resistor with a resistance of 19 kΩ, and a battery that has a potential difference of 170 V. What is the charge Q, in microcoulombs, on the capacitor when the current in the resistor equals one half its maximum value.
You charge an initially uncharged 69.1 mF capacitor through a 24.9 2 resistor by means of a 9.00 V battery having negligible internal resistance. Find the time constant t of the circuit. What is the charge Q on the capacitor 1.25 time constants after the circuit is closed? What is the charge Qo after a long amount of time has passed? C Qo
12. A lightbulb uses 4 A when it is connected across a 120 V source. What is the resistance in the resistane = V Vano -301 210-3012 lightbulb? 13. What is work? the transfer of energy by a force object agit is Displaced acting on by vitve opergy which a body possesses 14. What is kinetic energy? tion 15. What is potential energy? energy that is waiting to be a 16. A boy pushes on a parked car with a force of 111 N. The car does not move. How much work does the boy do on the car? Explain your answer. Work : Owork becasue the car. didn't move 17. In the absence of air resistance, how would the acceleration of a 2.0 kg book and the acceleration of a 3.3 kg rock differ if the objects were dropped from the same height? 18. What are Newton's three laws of motion (summarize them)? ject will not Law I an Change unle torce is acted on gecond low, the force On un object is equal! mass & its accelertion 3 law, when 2 bu interact they apply Forces to each oth 19. If your weight is 777 N, how much…
A RC circuit is constructed with a (initially uncharged) 5.00-µF capacitor and a 5.00- ohm resistor, which are connected in series to an ideal battery with emf 60.0 V. At a certain time after the circuit is connected, it is found that heat is being produced in the resistor at a rate of 300 W. How much energy is stored in the capacitor at that time (in Joules)? O None of these 1.8x104 1.13x10-3 3.75x10-3 9x10-3
The circuit schematic shows a combination of resistors connected to a battery. All resistances are given in Ohms. The voltage of the ideal battery is 32 Volts. (a) If all these resistors were replaced with a single equivalent resistor, what would its resistance be? (b) What is the current supplied by the battery? (c) What is the current through the 125 Ohm resistor? ww ww 225 435 ww 125 Vbatt 325 ww
In the figure the current in resistance 6 is i6 = 1.39 A and the resistances are R1 = R2 = R3 = 2.05 Q, R4 = 17.4 N, R5 = 8.80 2, and R6 = 4.742. What is the emf of the ideal battery? R R R Rg 16 卡