Three resistors of value 1.0 , 2.0 and 4.0 are connected in a parallel circuit with a 6.0 V battery.
a) What is the total equivalent resistance?
b) What is the across each resistor?
c) The power delivered to the 4.0 resistor?
1/Req = 1/1 + 1/2 + 1/4
1/Req = 4/4 + 2/4 + 1/4 = 7/4
so
Req = 4/7
The voltage is the same across all elements in parallel
i for the 4 ohm = 6/4 = 1.5 amps
P = i V or i^2 R = 1.5 * 6 or 2.25 * 4
= 9 watts
a) The total equivalent resistance is like trying to find a consensus at a family dinner. It's the resistance that all three resistors agree on. In this case, the formula for calculating the total equivalent resistance of resistors connected in parallel is 1/R_total = 1/R1 + 1/R2 + 1/R3. So let's do the math: 1/RT = 1/1.0 + 1/2.0 + 1/4.0 which gives you a total equivalent resistance of 0.5833 ohms.
b) Now, let's talk about the voltage across each resistor. Just like sharing pizza, the voltage will be divided among the resistors based on their resistance values. The formula for calculating the voltage across each resistor in a parallel circuit is V = (I * R), where V is the voltage, I is the current, and R is the resistance. Since the resistors are connected in parallel, they all have the same voltage across them. So each resistor will receive 6.0 volts.
c) To find the power delivered to the 4.0 ohm resistor, we can use the formula P = (V^2) / R, where P is the power, V is the voltage, and R is the resistance. Plugging in the numbers, we get P = (6.0^2) / 4.0, which gives us a power of 9.0 watts. That's enough power to make the 4.0 ohm resistor feel pretty hot under the collar!
To find the total equivalent resistance, we can use the formula for calculating the total resistance in a parallel circuit:
1/R_total = 1/R1 + 1/R2 + 1/R3
Substituting the given resistor values, we have:
1/R_total = 1/1.0 + 1/2.0 + 1/4.0
Simplifying this equation gives:
1/R_total = 0.5 + 0.25 + 0.125
1/R_total = 0.875
Taking the reciprocal of both sides, we find:
R_total = 1/0.875
R_total = 1.1429 ohms
Therefore, the total equivalent resistance is approximately 1.1429 ohms.
To find the voltages across each resistor, we can use Ohm's law:
V = I * R
Since the resistors are in parallel, the voltage across each resistor is the same as the battery voltage, which is 6.0V.
Therefore, the voltage across each resistor is 6.0V.
To find the power delivered to the 4.0 ohm resistor, we can use the formula:
P = (V^2) / R
Substituting the given values:
P = (6.0^2) / 4.0
P = 36 / 4
P = 9 watts
Therefore, the power delivered to the 4.0 ohm resistor is 9 watts.
To find the answers to these questions, we can use the laws and formulas of parallel circuits. Let's break it down step by step:
a) To find the total equivalent resistance in a parallel circuit, we use the formula:
1/Requivalent = 1/R1 + 1/R2 + 1/R3 + ...
In this case, R1 = 1.0 Ω, R2 = 2.0 Ω, and R3 = 4.0 Ω. Plugging these values into the formula, we get:
1/Requivalent = 1/1.0 + 1/2.0 + 1/4.0
Simplifying this equation, we find:
1/Requivalent = 2/4 + 1/2 + 1/4
Adding the fractions, we get:
1/Requivalent = 1 + 1/2 + 1/4
Combining like terms, we have:
1/Requivalent = 4/4 + 2/4 + 1/4
1/Requivalent = 7/4
Taking the reciprocal of both sides, we find:
Requivalent = 4/7 Ω
Therefore, the total equivalent resistance in this parallel circuit is 4/7 Ω.
b) To find the voltage across each resistor in a parallel circuit, we can use the fact that the voltage across all components connected in parallel is the same as the voltage across the battery. In this case, the battery voltage is 6.0 V.
Therefore, the voltage across each resistor in this circuit is 6.0 V.
c) The power delivered to a resistor can be calculated using the formula:
P = VI
where P is the power, V is the voltage across the resistor, and I is the current flowing through the resistor.
To find the power delivered to the 4.0 Ω resistor, we need to find the current flowing through it. For a parallel circuit, the total current is the sum of the currents through each resistor:
Itotal = I1 + I2 + I3 + ...
In this case, we can use Ohm's Law to find the current flowing through each resistor:
I1 = V/R1
I2 = V/R2
I3 = V/R3
Substituting the given values, we get:
I1 = 6.0 V / 1.0 Ω = 6.0 A
I2 = 6.0 V / 2.0 Ω = 3.0 A
I3 = 6.0 V / 4.0 Ω = 1.5 A
Now we can calculate the total current:
Itotal = I1 + I2 + I3 = 6.0 A + 3.0 A + 1.5 A = 10.5 A
Finally, we can calculate the power delivered to the 4.0 Ω resistor:
P4Ω = V × I4Ω = 6.0 V × 1.5 A = 9.0 W
Therefore, the power delivered to the 4.0 Ω resistor is 9.0 W.