Table of Contents

**D.C. Circuit**

The closed path followed by a direct current is called a d.c. circuit. A d.c. circuit essentially consists of a source of direct voltage (e.g. battery), the conductors used to carry current, and the load. a torch bulb (i.e. load) connected to a battery through conducting wires. The direct current starts from the positive terminal of the battery and comes back to the starting point via the load. The direct current follows the closed path ABCDA and hence ABCDA is a d.c. circuit. The load for a d.c. the circuit is usually resistant. In a d.c. circuit loads (i.e. resistances) may be connected in series or series-parallel.

**Direct Current**

The current that always flows in one direction is called **direct current **(d.c.). The current is supplied by a cell battery or d.c. the generator is direct current. (Fig.1.1 DC circuit), the battery supplies direct current to the bulb. The direction of the current is along ABCDA and it always flows in this direction. Note that direct current means steady direct current (i.e. one of constant magnitude) unless stated otherwise.

**Resistors in Series**

A number of resistors are said to be connected in series if the same current flows through each resistor and there is only one path for the current flow throughout. Consider three resistors of resistances R_{1}, R_{2}, and R_{3} connected in series across a battery of E volts as shown in fig. The total resistance Rt is given by ;

R_{T} = R_{1}+ R_{2} + R_{3},

Hence, when a number of resistances are connected in series, the total or equivalent resistance is equal to the sum of the individual resistances. Thus we can replace the series-connected resistor

shown in Fig. 2. with a single resistor, R_{T} = ( R_{1} + R_{2} + R_{3}). This will enable us to calculate the circuit current easily (I = E/R_{T}).

(1) R_{T} = R_{1}+ R_{2} + R_{3}

R_{T}/V_{2} = R_{1}/V^{2}+ R_{2}/V^{2} + R_{3}/V^{2}or 1/P_{T} = 1/P_{1}+ 1/P_{2}+ 1/P_{3}

where PT is the total power dissipated by the series circuit and P_{1}, P_{2} and P_{3} are the powers dissipated by individual resistors.

(2) The total conductance G_{T} of the circuit is

G_{T} = 1/R_{T} = 1/R_{1} + R_{2} + R_{3}

Also, 1/G_{T} = 1/G_{1} + 1/G_{2} + 1 /G_{3}

Here, G_{1} = 1/R_{1} ; G_{2} = 1/R_{2} ; G_{3 }= 1/R_{3}

The total power dissipated = I^{2 }R_{T} = I^{2}/G_{T }= E^{2}G_{T }

**Resistors in Parallel**

A number of resistors are said to be connected in parallel if the voltage across each resistor is the same and there are as many paths for current as the number of resistors. Consider three resistors of resistances R_{1}, R_{2} and R_{3} connected in parallel across a battery of E volts as shown in Fig. 1.3. Then total resistance RT is given by ;

1/R_{T}=1/R_{1}+ 1/R_{2} + 1/R_{3}

Hence, when a number of resistances are connected in parallel, the reciprocal of the total resistance is equal to the sum of reciprocals of individual resistances. Again, we can replace the parallel connected resistors shown in Fig. 1.3 with a single resistor RT.

(1) 1/R_{T}= 1/R_{1} + 1/R_{2} + 1/R_{3}

or V^{2}/R_{T}= V^{2}/R_{1} +V^{2}/R_{2}+ V^{2}/R_{3}

or P_{T} = P_{1}+ P_{2}+ P_{3}

where is the total power dissipated by the parallel circuit and P_{1}, P_{2} and P_{3 }are the powers dissipated by individual resistors.

(2) The total conductance GT of the circuit is

G_{T} = G_{1}+ G_{2}+ G_{3}

Here, G_{1} = 1/R_{1} ; G_{2}, = 1/R_{2} ; G_{3} = 1 /R_{3}

We can also express currents I_{1}, I_{2} , and I_{3 }in terms of conductances.

I_{1} = E/R_{1}

EG_{1}= IG_{1}/G_{T}

I *G_{1}/G_{T}

I * G1/ G1 + G2 + G3

Similarly,

I_{2} =I * G_{2}/G_{1} + G_{2} + G_{3} ; I_{3}= I * G_{3}/G_{1} + G_{2} + G_{3}

The total power dissipated,

P = E^{2}/R_{T } = E^{2}G_{T}