When 2 resistors, R1 and R2 are connected in series across a 6V battery the potential difference across R1 is 4V. When R1 and R2 are connected in parallel to the same battery the current through R2 is 0.45 A. Find the values of R1 and R2.
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Series Circuit.
V1 = 4Volts.
V2 = 6-4 = 2Volts.
R1 = 2R3.
Parallel Circuit.
R2 = V/I = 6/0.45 = 13.33 Ohms.
R1 = 2R2 = 2*13.33 = 26.66 Ohms.
Why did the resistor go on a diet? Because it wanted to have less resistance! Now, let's solve this electrical puzzle.
Let's start with the series connection first. Since the potential difference across R1 is 4V, it means that the potential difference across R2 is 6V - 4V = 2V. Are you following along?
Now, let's move on to the parallel connection. We know that the current through R2 is 0.45A. So, using Ohm's Law, we can determine the resistance of R2.
Ohm's Law states that V = IR, where V is the potential difference, I is the current, and R is the resistance. In this case, V is 2V and I is 0.45A. Therefore, we can calculate R2 as R2 = V/I = 2V/0.45A.
Now, to find R1, we can use the fact that the potential difference across R1 in the series connection is 4V. Using Ohm's Law again, we can determine R1 as R1 = V/I = 4V/0.45A.
So, the values of R1 and R2 are R1 = 8.89 ohms and R2 = 4.44 ohms.
Voilà! The resistor values have been revealed. Hope this illuminating response sparked some laughter!
To find the values of R1 and R2, we can use the following equations:
1. When two resistors are connected in series:
Total voltage (Vt) = Voltage across R1 (V1) + Voltage across R2 (V2)
2. When two resistors are connected in parallel:
Total current (It) = Current through R1 (I1) + Current through R2 (I2)
Given information:
- Potential difference across R1 (V1) = 4V
- Total voltage (Vt) = 6V
- Current through R2 (I2) = 0.45A
Step 1: Calculation using series connection:
Since R1 and R2 are connected in series, the total voltage is divided between them.
Vt = V1 + V2
Substituting the given values:
6V = 4V + V2
V2 = 6V - 4V
V2 = 2V
Step 2: Calculation using parallel connection:
Since R1 and R2 are now connected in parallel, the current is divided between them.
It = I1 + I2
Substituting the given values:
I2 = 0.45A
It = I1 + 0.45A
Since It = Vt / Rtotal and the total resistance is equal to the reciprocal of the sum of the reciprocals of R1 and R2:
It = Vt / (1/R1 + 1/R2)
Substituting the given values:
I1 + 0.45A = 6V / (1/R1 + 1/R2)
Step 3: Solving the equations simultaneously:
We have two equations:
1. V2 = 2V
2. I1 + 0.45A = 6V / (1/R1 + 1/R2)
From equation 1, we know that V2 = 2V. Therefore, R2 must be equal to V2 / I2:
R2 = V2 / I2
R2 = 2V / 0.45A
R2 = 4.44 ohms (rounded to two decimal places)
To find R1, we can substitute the given values into equation 2 and solve for R1:
I1 + 0.45A = 6V / (1/R1 + 1/4.44)
Simplifying further:
I1 + 0.45A = 6V / ((4.44 + R1) / (R1 * 4.44))
Cross-multiplication:
(I1 + 0.45A) * (R1 * 4.44) = 6V * (4.44 + R1)
4.44 * I1 * R1 + 0.45A * R1 * 4.44 = 26.64V + 6V * R1
4.44 * I1 * R1 - 6V * R1 = 26.64V - 0.45A * R1 * 4.44
(4.44 * I1 - 6V) * R1 = 26.64V
R1 = 26.64V / (4.44 * I1 - 6V)
Substituting the given value for I2:
R1 = 26.64V / (4.44 * I1 - 6V)
R1 = 26.64V / (4.44 * 0.45A - 6V)
R1 = 19.3 ohms (rounded to one decimal place)
Therefore, the values of R1 and R2 are approximately:
R1 = 19.3 ohms
R2 = 4.44 ohms
To solve this problem, we can apply the principles of Ohm's law and the laws of series and parallel resistors.
Let's start by analyzing the series connection. In this case, the potential difference across each resistor adds up to the total potential difference across the battery. Since the potential difference across R1 is given as 4V and the total potential difference across the battery is 6V, we can conclude that the potential difference across R2 is 6V - 4V = 2V.
Using Ohm's law (V = I * R), where V is the potential difference, I is the current, and R is the resistance, we can find the values of R1 and R2.
For the series connection:
V = 4V and I = ?
We don't have the value of the current, but since the resistors are connected in series, the current is the same through both resistors. We will use this information later.
For the parallel connection:
I = 0.45A, V = 2V, and R2 = ?
Again, since the resistors are connected in parallel, the potential difference across each resistor is the same. Therefore, the current through R2 is also equal to the total current.
Now, let's find the value of R1 using the information we have about the series connection. We can use Ohm's law: 4V = I * R1.
Since the current is the same in both connections, we can equate the value of I obtained from the parallel connection to the value obtained from the series connection: I = 0.45A = 4V / R1.
Simplifying the equation, we have 0.45A = 4V / R1.
Now, let's solve for R1:
Cross multiplying, we get 0.45A * R1 = 4V.
Dividing both sides by 0.45A, we have R1 = 4V / 0.45A.
Calculating R1, we find R1 = 8.89 Ω.
Finally, to find R2, we can use Ohm's law in the parallel connection: 2V = 0.45A * R2.
Simplifying the equation, we have 2V = 0.45A * R2.
Now, let's solve for R2:
Dividing both sides by 0.45A, we have R2 = 2V / 0.45A.
Calculating R2, we find R2 = 4.44 Ω.
Thus, the values of R1 and R2 are R1 = 8.89 Ω and R2 = 4.44 Ω, respectively.