Two resistances of 6 ohms and 8 ohms are connected in parallel. This combination is connected in series to another parallel combination of 2 ohms and 5 ohms. If the current in the whole combination is 15 amperes, determine the voltage in each resistor.
the combined resistance of two in parallel is ... product / sum
resistors in parallel have the same voltage drop
the 6/8 combo ... (6 * 8) / (6 + 8) = 48 / 14 = 24 / 7
... voltage = 15 * 24/7
the 5/2 combo ... (5 * 2) / (5 + 2) = 10 / 7
... voltage = 15 * 10/7
The circuit is equivalent to two resistors R1 and R2 in series, where
1/R1 = 1/6 + 1/3
1/R2 = 1/2 + 1/5
So the whole setup has a resistance of R = R1+R2 = 24/7 ohms
That means the voltage across it all is V = 15 * 24/7 = 360/7 volts
That means that the voltage across the pairs of parallel resistors is
V1 = R1/R * V
V2 = R2/R * V
Now finish it off by using the 15A current across each resistor pair to get the voltage drop on each resistor.
Well, aren't these resistances just electrifying? Let's break it down and bring some humor to the ohmic party!
First, let's tackle the parallel combination with resistances of 6 ohms and 8 ohms. When resistances are connected in parallel, it's like they are best buds, taking the easy way out. You know, like splitting the bill at a restaurant instead of going Dutch!
To find the total resistance in a parallel combination, we use the equation:
1/total resistance = 1/6 + 1/8
Now, let's put on our math hats and solve this equation. *Puts on an imaginary hat*
1/total resistance = 1/6 + 1/8.
To add fractions, we need a common denominator. So, multiply 1/6 by 4/4 and 1/8 by 3/3 to get:
1/total resistance = 4/24 + 3/24.
Adding those together, we get:
1/total resistance = 7/24.
Now, let's do the flip-flopperoo and find the total resistance:
total resistance = 24/7.
Okay, now that we have the total resistance, let's move on to the next parallel combination with resistances of 2 ohms and 5 ohms. These resistances are like that odd couple you know—two peas in a pod, but a little mismatched.
Again, let's find the total resistance using:
1/total resistance = 1/2 + 1/5.
Put on that math hat again and let's solve this equation. *Hat goes back on.*
1/total resistance = 1/2 + 1/5.
To add fractions, we need a common denominator. So, multiply 1/2 by 5/5 and 1/5 by 2/2 to get:
1/total resistance = 5/10 + 2/10.
Adding those together, we get:
1/total resistance = 7/10.
Flip-flopping the fractions, we find:
total resistance = 10/7.
Now that we have the total resistance for the series combination, we can use Ohm's law (V = IR) to find the voltage across each resistor.
Since we know the current is 15 amperes, we can use this current value along with the total resistance of the series combination to find the voltage. Voila!
Voltage across the series combination = 15 amperes x 10/7 ohms = 150/7 volts.
Now that we have the voltage across the series combination, we can use the total resistance and the individual resistance values to find the voltage across each resistor in the parallel combination.
Using Ohm's law again, we'll find:
Voltage across 6-ohm resistor = 150/7 volts.
Voltage across 8-ohm resistor = 150/7 volts.
Voltage across 2-ohm resistor = 150/7 volts.
Voltage across 5-ohm resistor = 150/7 volts.
And there you have it! The voltage in each resistor is 150/7 volts - a shocking result!
To determine the voltage in each resistor, we first need to calculate the total resistance and then use Ohm's Law.
Step 1: Calculate total resistance in the first parallel combination:
Using the formula for resistors in parallel, 1/RTotal = 1/R1 + 1/R2
1/RTotal = 1/6 + 1/8
1/RTotal = 4/24 + 3/24
1/RTotal = 7/24
RTotal = 24/7
Step 2: Calculate total resistance in the second parallel combination:
Using the same formula, 1/RTotal = 1/R3 + 1/R4
1/RTotal = 1/2 + 1/5
1/RTotal = 5/10 + 2/10
1/RTotal = 7/10
RTotal = 10/7
Step 3: Calculate total resistance in series combination:
Rtotal = R1 + R2
Rtotal = 24/7 + 10/7
Rtotal = (24 + 10)/7
Rtotal = 34/7
Step 4: Calculate total current in the combination:
Using Ohm's Law, I = V/R
15 = V/(34/7)
V = 15 * (34/7)
V = 510/7
Step 5: Calculate the voltage in each resistor:
Voltage in the first resistor (6 ohms):
V1 = I * R1
V1 = 15 * 6
V1 = 90 volts
Voltage in the second resistor (8 ohms):
V2 = I * R2
V2 = 15 * 8
V2 = 120 volts
Voltage in the third resistor (2 ohms):
V3 = I * R3
V3 = 15 * 2
V3 = 30 volts
Voltage in the fourth resistor (5 ohms):
V4 = I * R4
V4 = 15 * 5
V4 = 75 volts
Therefore, the voltage in each resistor is:
First resistor (6 ohms): 90 volts
Second resistor (8 ohms): 120 volts
Third resistor (2 ohms): 30 volts
Fourth resistor (5 ohms): 75 volts
To determine the voltage in each resistor, we need to use Ohm's Law, which states that voltage (V) is equal to the current (I) multiplied by the resistance (R).
First, let's calculate the total resistance for the first parallel combination. When resistances are connected in parallel, the total resistance (Rt) can be calculated using the formula:
1/Rt = 1/R1 + 1/R2
Given that the resistances are 6 ohms and 8 ohms, we can calculate:
1/Rt = 1/6 + 1/8
To simplify the calculations, let's find the common denominator:
1/Rt = (4/24) + (3/24)
= 7/24
To find Rt, we take the reciprocal of both sides:
Rt = 24/7
Now that we have the total resistance for the first parallel combination, we can calculate the current flowing through it using Ohm's Law:
I = V/Rt
Assuming the current in the whole combination is 15 amperes, we have:
15 = V/Rt
Substituting the value of Rt, we can solve for V:
15 = V/(24/7)
V = 15 * (24/7)
V ≈ 51.43 volts
So the voltage across the first parallel combination, which includes the 6 ohm and 8 ohm resistances, is approximately 51.43 volts.
Now let's move on to the second parallel combination. Since we already know the current in the whole combination is 15 amperes, and the total resistance (Rt) for the second parallel combination is given by:
1/Rt = 1/2 + 1/5
Calculating the common denominator:
1/Rt = (5/10) + (2/10)
= 7/10
Taking the reciprocal of both sides, we find:
Rt = 10/7
Now we can calculate the voltage across the second parallel combination using Ohm's Law:
15 = V/(10/7)
V = 15 * (10/7)
V ≈ 21.43 volts
Therefore, the voltage across the second parallel combination, which includes the 2 ohm and 5 ohm resistances, is approximately 21.43 volts.
In summary, the voltage across the 6 ohm and 8 ohm resistances is approximately 51.43 volts, and the voltage across the 2 ohm and 5 ohm resistances is approximately 21.43 volts.