A resistance r1=12ohms is connected in series with another resistance r2= 8 ohms. What resistance r3 must be connected in parallel with the combination of r1 and r2 so that the equivalent resistance is equal to the resistance r1?
r1 = [(r1 + r2) * r3] / (r1 + r2 + r3)
plug in values and solve for r3
Given:
R1 = 12 ohms.
R2 = 8 ohms.
R1 + R2 = 12 + 8 = 20 ohms.
(R3*20)/(R3+20) = R1.
(R3*20)/(R3+20) = 12.
20R3 = 12R3 + 240,
R3 = ?
To find the resistance r3 that must be connected in parallel with the combination of r1 and r2, we need to calculate the equivalent resistance of the series combination of r1 and r2 and then determine the value of r3 that would make it equal to r1.
Step 1: Calculate the equivalent resistance of the series combination of r1 and r2.
When resistors are connected in series, their resistances add up. So, the equivalent resistance of the series combination of r1 and r2 is given by:
Req = r1 + r2
= 12 ohms + 8 ohms
= 20 ohms
Step 2: Determine the value of r3 that would make the equivalent resistance equal to r1.
Since we want the equivalent resistance to be equal to r1 (which is 12 ohms), we need to find the value of r3 such that it, in parallel with the combination of r1 and r2, gives an equivalent resistance of 12 ohms.
The formula for calculating the equivalent resistance of resistors connected in parallel is given by:
1/Req = 1/r1 + 1/r3
Rearranging the equation to solve for r3, we have:
1/r3 = 1/Req - 1/r1
Substituting the given values, we get:
1/r3 = 1/20 ohms - 1/12 ohms
To simplify this expression, we need to find a common denominator:
1/r3 = (12 - 20)/(20 * 12)
= -8/240
= (-1/30) ohms
Taking the reciprocal of both sides, we have:
r3 = -30 ohms
Since resistance cannot be negative, we conclude that there is no value of r3 that would make the equivalent resistance equal to r1.