#1. Ten Christmas lights are connected in series and plugged into a 120 V wall outlet. Each light is identical and can be thought of as a resistor.

a) What should the value of this resistance be if the total string is to dissipate 50 W?

b) How much power is dissipated in each resistor?

c) How much current flows in each resistor?

#2. Ten Christmas lights are now connected in parallel. Repeat problem #1 for this new design.

#3. Which string lights (#1 or #2) makes a better consumer product? Why?

Sure, let's go through each question step by step:

1a) To find the value of resistance for the series-connected Christmas lights, we need to use Ohm's Law. Ohm's Law states that the current flowing through a resistor is equal to the voltage across it divided by its resistance. In this case, the total power dissipated by the string is given as 50 W, and the total voltage is 120 V.

Using the formula for power (P) in terms of current (I) and resistance (R): P = I^2 * R, we can rearrange the equation to solve for resistance (R): R = P / I^2.

Since the lights are connected in series, the current (I) passing through each light is the same. Therefore, we can find the current by dividing the total voltage (120 V) by the number of lights (10) using Ohm's Law: I = V / N, where V is the voltage and N is the number of lights.

Now, we can calculate resistance: R = P / I^2 = 50 W / (120 V / 10)^2.

1b) In a series circuit, the current remains the same throughout, so the power dissipated in each resistor (light) is the total power divided by the number of lights: P_resistor = P_total / N = 50 W / 10.

1c) Since the same current flows through each resistor in a series circuit, the current is equal to the total current divided by the number of lights: I_resistor = I_total / N = (120 V / 10) / 10.

Now, let's move on to the second problem:

2) When the lights are connected in parallel, the voltage across each light remains the same. Therefore, the total voltage (V_total) is still 120 V.

2a) We can calculate the resistance for the parallel-connected Christmas lights using Ohm's Law. In a parallel circuit, the total current (I_total) is the sum of the currents flowing through each light (I_resistor): I_total = I_1 + I_2 + ... + I_N.

Since the voltage across each light is the same (120 V), we can calculate the current flowing through each light using Ohm's Law: I_resistor = V / R, where V is the voltage and R is the resistance.

The total resistance for a parallel circuit (R_total) is given by the reciprocal of the sum of the reciprocals of individual resistances: 1/R_total = 1/R_1 + 1/R_2 + ... + 1/R_N.

Now, we can rearrange the equation to solve for R_total: R_total = 1 / (1/R_1 + 1/R_2 + ... + 1/R_N).

2b) In a parallel circuit, the voltage across each resistor (light) is the same as the total voltage. Therefore, the power dissipated in each resistor is given by the formula: P_resistor = V^2 / R.

2c) In a parallel circuit, the current flowing through each resistor varies. We can calculate the individual currents using Ohm's Law: I_resistor = V / R.

Finally, let's address the last question:

3) To determine which string lights make a better consumer product, we need to consider factors such as safety, durability, energy efficiency, and convenience.

- In terms of safety, both series and parallel connections have their advantages. In a series connection, if one light burns out, the entire string can go out. This can help prevent overheating. In a parallel connection, if one light burns out, the rest remain lit, ensuring that the user does not lose the entire functionality of the string.

- For durability, it can vary depending on the specific product. However, with series connections, if one light fails, it can affect the functionality of the entire string. In contrast, with parallel connections, if one light fails, it will not affect the others.

- Energy efficiency can be better in the parallel connection, as each light has its own pathway for current flow and can be independently controlled. In a series connection, the voltage across each light adds up, which might not be as energy-efficient.

- In terms of convenience, parallel connections offer more flexibility. With independent control of each light, if one goes out, it is easier to identify and replace. In a series connection, troubleshooting can be more time-consuming.

Considering these factors, the better consumer product might depend on the specific requirements and preferences of the user.

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