Answer to Question #172174 in Electric Circuits for JohnDoe

Question

2.8 Connect the same resistors in parallel. (a) Make a schematic diagram. Then, find (b)

the current and voltage in each load and (c) the total Current, Resistance and Voltage in the circuit.

Answer 1

Let,

A circuit given with:

• 3 resistors R₁ R₂ and R₃ (given : R₁ =R₂ = R₃=R $\Omega$ )
• voltage V

a)arranged in parallel combination as given below (fig.1)$\to$

b) $\implies$ As fig.1 shows resistors in parallel, wired to a voltage source. Resistors are in parallel when each resistor is connected directly to the voltage source by connecting wires having negligible resistance.

Thus,

$\to$ each resistor has the full voltage of the source applied to it, i.e, V volts.

$\implies$Since each resistor in the circuit has the full voltage , so according to OHM'S LAW the current following through the individual resistors are :

I₁=${V\over R_{1}}$ I₂=${V\over R_{2}}$ I₃=${V\over R_{3}}$ ........(1)

$\boxed{\therefore I_1=I_2= I_3={V\over R_{}}}$

c) TOTAL CURRENT IN CIRCUIT

According to LAW OF CONSERVATION OF CHARGE the current I produced by the source is the sum of the currents in all the loads.

$\therefore$ $\boxed{I = I_1 + I_2 + I_3 }$ _.......(2)

substituting the expression (1) in (2) gives :

I=${V\over R_{1}}$ + ${V\over R_{2}}$ + ${V\over R_{3}}$

I= V (${1\over R_{1}}$ + ${1\over R_{2}}$ + ${1\over R_{3}}$ ) .......(3)

note that: OHM's law for equivalent single resistence (fig.1) gives:

I=${V\over R_{P}} =V({1\over R_{P}} )$ .......(4)

TOTAL RESISTENCE IN CIRCUIT

$\implies$ using (3) and (4) we get :

$\boxed{{1\over R_{P}}={1\over R_{1}}+{1\over R_{2}}+{1\over R_{3}}}$ .........(5)

$\therefore$ total resistance in circuit is R_{P .}

TOTAL VOLTAGE IN CIRCUIT

Total voltage in circuit is the applied voltage of V volts.

and according to OHM'S law:

$\boxed{V=IR_P}$

NOTE: Concepts used:

1: Ohm's Law

2: conservation of current

and Kirchhoff's law is also useful in deriving equations.