Question

For a circuit in series, why does V1+V2+V3=V total. For a circuit in parallel, why does I1+I2+I3= I total?

Answer 1

In the series circuit, Since there is no alternative path for currents, So whole current must pass through each each resistor, which means

I total = I1 = I2 = I3

Now Using Ohm's law:

Vtotal = I(total)*Req

In series Req = R1 + R2 + R3

Vtotal = I(total)*(R1 + R2 + R3)

Also in each resistor must follow Ohm's law, So

V1 = i1*R1

V2 = i2*R2

V3 = i3*R3

Add All equation

V1 + V2 + V3 = i1*R1 + i2*R2 + i3*R3

Since i1 = i2 = i3 = i(total)

V1 + V2 + V3 = i(total)*(R1 + R2 + R3)

fromBold equation

V1 + V2 + V3 = Vtotal

Part B.

Since in parallel circuit, each resistor is directly connected to battery means there is no internal/external resistance between battery and resistor, So

V1 = V2 = V3 = V

Current through each resistor will be

i1 = V1/R1

i2 = V2/R2

i3 = V3/R3

Add all equations

i1 + i2 + i3 = V1/R1 + V2/R2 + V3/R3

Since V1 = V2 = V3 = V

i1 + i2 + i3 = V*(1/R1 + 1/R2 + 1/R3)

Since in parallel circuit

1/Req = 1/R1 + 1/R2 + 1/R3

i1 + i2 + i3 = V/Req

So, from Ohm's law:

V = i(total)*Req

i1 + i2 + i3 = i(total)