# Current algebra

#### calculator20

1. Homework Statement
An electric circuit consists of 3 identical resistors of resistance R connected to a cell of emf E and negligible internal resistance. What is the magnitude of the current in the cell? (in the diagram two of the resistors are in parallel with each other then the other in series with that pair)

2. Homework Equations
I = V/R

1/R = 1/r + 1/r +...

R = r1 + r2

3. The Attempt at a Solution

It's a multiple choice question and the answer is 2E/3R but I can't see where that has come from.

If I said that R was 2 ohms then the combined resistance would be 3 ohms but that doesn't help. I can't see where the 2/3 has come from please help, thank you

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#### gneill

Mentor
Evaluate the total resistance symbolically.

#### BOAS

1. Homework Statement
An electric circuit consists of 3 identical resistors of resistance R connected to a cell of emf E and negligible internal resistance. What is the magnitude of the current in the cell? (in the diagram two of the resistors are in parallel with each other then the other in series with that pair)

2. Homework Equations
I = V/R

1/R = 1/r + 1/r +...

R = r1 + r2

3. The Attempt at a Solution

It's a multiple choice question and the answer is 2E/3R but I can't see where that has come from.

If I said that R was 2 ohms then the combined resistance would be 3 ohms but that doesn't help. I can't see where the 2/3 has come from please help, thank you

Can you calculate what the equivalent resistance of this circuit is?

#### calculator20

I think it's (R1xR2)/(R1+R2) for the parallel part then (R1xR2)/(R1+R2) + R to add the series part in but don't know how to combine those?

#### BOAS

I think it's (R1xR2)/(R1+R2) for the parallel part then (R1xR2)/(R1+R2) + R to add the series part in but don't know how to combine those?
In order to add fractions, you need to have the same denominator in both.

and remember that all three resistors are identical. This will come in handy when simplifying your result.

#### calculator20

Ok so ((RxR)/(R+R)) + R

(2R/2R) + R? I don't think that's right but don't understand how I would get the same denominator either

#### BOAS

Ok so ((RxR)/(R+R)) + R

(2R/2R) + R? I don't think that's right but don't understand how I would get the same denominator either
Not quite.

$\frac{R \times R}{R + R} + R = \frac{R^2}{2R} + R$

If you were asked $\frac{1}{2} + 1 = ?$, you would need to recognise that this is equivalent to $\frac{1}{2} + \frac{2}{2} = \frac{1 + 2}{2}$.

You need to use this very same trick.

#### calculator20

1/2 R2/R +R/R

Do I have to separate the square now? Sorry, still can't see how I'm going to get the 2/3. Really appreciate your help so far

(1/2R2 + R)/R?

#### BOAS

1/2 R2/R +R/R

Do I have to separate the square now? Sorry, still can't see how I'm going to get the 2/3. Really appreciate your help so far
$\frac{R^2}{2R} + R$

If you simplify this expression, what do you get?

Edit - Sorry, this is my final edit. I misread what you typed.

Last edited:

R2/2R?

#### BOAS

If you cancel the $R$ terms in the fraction, you get $\frac{R}{2} + R = \frac{3R}{2}$

#### calculator20

Thank you so much that makes sense!

#### BOAS

Thank you so much that makes sense!
You're welcome.

To recap (so you have the entire argument in one place):

Two resistors in parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$.
The equivalent resistance of this pair is therefore $\frac{R_1 \times R_2}{R_1 + R_2}$

Two resistors in series: $R_{eq} = R_1 + R_2$

In your situation, you treat the equivalent resistance of the parallel setup, as a single resistor in series with your third resistor.

$R_{total} = \frac{R_1 \times R_2}{R_1 + R_2} + R_3$

Since $R_1 = R_2 = R_3$

$R_{total} = \frac{R^2}{2R} + R = \frac{R}{2} + R = \frac{3R}{2}$

and then it is just the case of applying ohm's law.

"Current algebra"

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