A circuit contains six 270 lamps and a 18.0 heater connected in parallel. The voltage across the circuit is 120 V.

(a) What is the current in the circuit when four lamps are turned on?
(b) What is the current in the circuit when all lamps are on?
(c) What is the current in the circuit when six lamps and the heater are operating?

R(Lp) = 270 Ohms each?

R(Htr) = 18 Ohms?

a. I = 4 * (120/270) = 1.78A.

b. I = 6 * (120/270) = 2.67A.

c. I = 2.67 + (120/18) = 9.34A
.

To solve this problem, we need to first understand the concept of parallel circuits. In a parallel circuit, the voltage across each component is the same, while the current through each component can vary.

(a) To find the current in the circuit when four lamps are turned on, we need to find the equivalent resistance and then use Ohm's Law (I = V/R) to calculate the current.

1. Find the equivalent resistance (R_eq) of four lamps:
Since the lamps are connected in parallel, the reciprocal of the equivalent resistance is equal to the sum of the reciprocals of individual resistances:
1/R_eq = 1/R1 + 1/R2 + 1/R3 + 1/R4
R_eq = 1 / (1/R1 + 1/R2 + 1/R3 + 1/R4)
R_eq = 1 / (1/270 + 1/270 + 1/270 + 1/270)
= 270 / 4
= 67.5 Ω

2. Use Ohm's Law to calculate the current:
I = V / R_eq
I = 120 V / 67.5 Ω
I ≈ 1.78 A

Therefore, the current in the circuit when four lamps are turned on is approximately 1.78 A.

(b) To find the current in the circuit when all lamps are on, we need to calculate the equivalent resistance using the same steps as in part (a) for six lamps:

1. Find the equivalent resistance (R_eq) of six lamps:
R_eq = 1 / (1/R1 + 1/R2 + 1/R3 + 1/R4 + 1/R5 + 1/R6)
= 1 / (1/270 + 1/270 + 1/270 + 1/270 + 1/270 + 1/270)
= 270 / 6
= 45 Ω

2. Use Ohm's Law to calculate the current:
I = V / R_eq
I = 120 V / 45 Ω
I ≈ 2.67 A

Therefore, the current in the circuit when all lamps are on is approximately 2.67 A.

(c) To find the current in the circuit when six lamps and the heater are operating, we need to calculate the equivalent resistance using the following steps:

1. Find the equivalent resistance (R_eq) of six lamps and the heater:
R_eq = 1 / (1/R1 + 1/R2 + 1/R3 + 1/R4 + 1/R5 + 1/R6 + 1/R_heater)
= 1 / (1/270 + 1/270 + 1/270 + 1/270 + 1/270 + 1/270 + 1/18)
= 270 / 6 + 15
≈ 60 Ω

2. Use Ohm's Law to calculate the current:
I = V / R_eq
I = 120 V / 60 Ω
I = 2 A

Therefore, the current in the circuit when six lamps and the heater are operating is 2 A.

To find the current in the circuit, we can use Ohm's Law, which states that current (I) is equal to voltage (V) divided by resistance (R). In this case, the resistance is determined by the lamps and the heater.

(a) When four lamps are turned on, the resistance will change. Let's assume that each lamp has a resistance of R_lamp. Since the lamps are connected in parallel, the total resistance of the lamps can be calculated as:
R_total_lamps = R_lamp / number_of_lamps_on

So, for this case, the total resistance of the lamps would be:
R_total_lamps = R_lamp / 4

We can find the resistance by using the formula:
R = V / I

Since V = 120 V, we can rearrange the formula to solve for current (I):
I = V / R

(b) When all lamps are on, the total resistance of the lamps would be:
R_total_lamps = R_lamp / 6

(c) When all six lamps and the heater are operating, the total resistance of the circuit can be calculated as follows:
R_total = R_total_lamps + R_heater

Now let's calculate the values step by step.

Step 1: Find the resistance of each lamp.
We need the resistance of each lamp (R_lamp) to calculate the total resistance of the lamps (R_total_lamps).

Step 2: Calculate the total resistance of the lamps when four lamps are turned on.
Use the formula R_total_lamps = R_lamp / 4 to find the total resistance of the lamps in this case.

Step 3: Calculate the current (I) when four lamps are turned on.
Use the formula I = V / R_total_lamps, where V is the voltage across the circuit (120 V) and R_total_lamps is the total resistance of the lamps.

Step 4: Calculate the total resistance of the lamps when all lamps are on.
Use the formula R_total_lamps = R_lamp / 6 to find the total resistance of the lamps in this case.

Step 5: Calculate the current (I) when all lamps are on.
Use the formula I = V / R_total_lamps, where V is the voltage across the circuit (120 V) and R_total_lamps is the total resistance of the lamps.

Step 6: Calculate the total resistance of the circuit when all six lamps and the heater are operating.
Add the total resistance of the lamps (R_total_lamps) to the resistance of the heater (R_heater) to find the total resistance of the circuit.

Step 7: Calculate the current (I) when all six lamps and the heater are operating.
Use the formula I = V / R_total, where V is the voltage across the circuit (120 V) and R_total is the total resistance of the circuit.