Four resistors each of resistance 4 Ù are connected to form a square loop. What is the equivalent resistance between two points on its diagonal AC or BD?
A-------------D
| |
B-------------C
That's what it looks like on my homework. Thank you in advance
Is this correct?
To find the equivalent resistance between points A and C (or B and D) on the diagonal, we can first analyze the symmetry of the circuit. Since the square loop is symmetrical with equal resistors, the resistance between A and C will be the same as the resistance between B and D.
Now, let's label the points on the square loop as follows:
A-------------D
| |
B-------------C
Next, let's find the total resistance for one side of the square loop.
Resistance between A and B:
Since the two resistors in parallel each have a resistance of 4 Ω, the total resistance across these two resistors is given by:
1 / R_ab = 1 / 4 Ω + 1 / 4 Ω
= 1 / (1/4 + 1/4) Ω
= 1 / (2/4) Ω
= 1 / (1/2) Ω
= 2 Ω
Now, we will consider the resistance between A and C (or B and D) on the diagonal.
Resistance between A and D (or B and C):
Since the two resistors have resistances of 2 Ω each (which we found earlier), and they are connected in series, the total resistance across these two resistors is simply the sum of their resistances:
R_ad = 2 Ω + 2 Ω
= 4 Ω
Hence, the equivalent resistance between points A and C (or B and D) on the diagonal is 4 Ω.
To find the equivalent resistance between the points on the diagonal AC or BD, we can use the concept of series and parallel resistors.
First, let's label the resistors in the square loop as R1, R2, R3, and R4. Each resistor has a resistance of 4 Ω.
To find the equivalent resistance between AC or BD, we need to consider the resistors that are connected in parallel along the diagonal.
If we look at diagonal AC, we can see that resistors R1 and R3 are connected in parallel, while resistors R2 and R4 are also connected in parallel.
To calculate the equivalent resistance between AC, we can use the formula for resistors connected in parallel:
1/Req = 1/R1 + 1/R3
Substituting the values, we get:
1/Req = 1/4Ω + 1/4Ω or 1/Req = 1/2Ω
Simplifying, we find:
1/Req = 1/(4Ω/2) or 1/Req = 2/4
Therefore, the equivalent resistance between AC is:
Req = 4Ω/2 or Req = 2Ω
Similarly, for diagonal BD, resistors R2 and R4 are in parallel. So the equivalent resistance between BD is also 2 Ω.
Hence, the equivalent resistance between two points on the diagonal AC or BD is 2 Ω.
R1=R(AD)+R(DC)=4+4= 8Ω
R2=R(AB)+R(BC)= 4+4= 8Ω
1/R =1/R1 +1/R2= 1/8 + 1/8= 2/8= 1/4
R= 4 Ω