Resistors in combination: Two identical resistors of resistance R = 24 Ω and a variable resistor R_x are connected to an ideal battery of voltage V as shown in the figure. What should be the value of the variable resistance Rx to make the voltage across the two parallel resistors equal to . A) 4.0 Ω B) 24 Ω C) 8.0 Ω D) 16 Ω E) 40 Ω

Question

Quizplus · Accepted Answer

First, we need to find the equivalent resistance of the two parallel resistors. Since they are identical, their equivalent resistance can be found using the formula:

Req = R/2

Where R is the resistance of one of the resistors.

Substituting R = 24 Ω, we get:

Req = 24/2 = 12 Ω

Now, we can find the current flowing through the two parallel resistors using Ohm's law:

I = V/Req

Substituting V = 11.67 V and Req = 12 Ω, we get:

I = 11.67/12 = 0.9725 A

Since the two parallel resistors are identical, the current will split equally between them. Therefore, the current flowing through each resistor can be found using Ohm's law:

I = V/R

Substituting V = 11.67 V and R = 24 Ω, we get:

0.9725 = 11.67/24

Solving for R, we get:

R = 11.67/0.9725 = 12 Ω

This means that the value of the variable resistor must also be 12 Ω in order to make the voltage across the two parallel resistors equal to 11.67 V. However, none of the answer choices matches this value. The closest choice is C) 8.0 Ω. We can check if this choice is valid by recomputing the current flowing through the parallel resistors using the new equivalent resistance:

Req = (R/2) + Rx

Substituting R = 24 Ω and Rx = 8.0 Ω, we get:

Req = (24/2) + 8.0 = 20 Ω

I = V/Req

Substituting V = 11.67 V and Req = 20 Ω, we get:

I = 11.67/20 = 0.5835 A

The current flowing through each resistor can be found using Ohm's law:

I = V/R

For the two parallel resistors, we have:

0.5835 = V1/24

V1 = 0.5835 * 24 = 14.004 V

And since the voltage across the variable resistor must also be 14.004 V, we have:

V2 = V - V1 = 11.67 - 14.004 = -2.334 V

This means that the direction of the current through the variable resistor is opposite to that of the battery, which is not physically possible. Therefore, C) 8.0 Ω is not a valid choice.

The only other possible choice is A) 4.0 Ω. We can check if this choice is valid by recomputing the current flowing through the parallel resistors using the new equivalent resistance:

Req = (R/2) + Rx

Substituting R = 24 Ω and Rx = 4.0 Ω, we get:

Req = (24/2) + 4.0 = 16 Ω

I = V/Req

Substituting V = 11.67 V and Req = 16 Ω, we get:

I = 11.67/16 = 0.7294 A

The current flowing through each resistor can be found using Ohm's law:

I = V/R

For the two parallel resistors, we have:

0.7294 = V1/24

V1 = 0.7294 * 24 = 17.504 V

And since the voltage across the variable resistor must also be 17.504 V, we have:

V2 = V - V1 = 11.67 - 17.504 = -5.834 V

This means that the direction of the current through the variable resistor is opposite to that of the battery, which is not physically possible. Therefore, A) 4.0 Ω is also not a valid choice.

In conclusion, there is no valid answer choice for this problem. The correct value of the variable resistor to make the voltage across the two parallel resistors equal to 11.67 V is 12 Ω.

Resistors in Combination: Two Identical Resistors of Resistance R =