Combination Of Resistors And Charging, Discharging Of Capacitors, JEE 2024

Resistors : The resistance of a conductor is the opposition that the conductor offers to the flow of charge. In this article, we will discuss the equivalent resistance in different combinations of resistors and different parameters during the charging and discharging of capacitors in series RC circuit.

Kirchhoff’s Current Law (Junction Law): It states that “The algebraic sum of currents meeting at a point of circuit is zero”. That means total current entering a junction equals the total current leaving the junction. This law is based on law of conservation of charge.

Kirchhoff’s Voltage Law (Junction Law)

Kirchhoff’s Voltage Law (Junction Law): It states that “The algebraic sum of all the potential differences along a closed loop is zero”.

i.e.,

While traversing a loop if potential increases, put a positive sign in expression and if potential decreases put a negative sign.

Combination Of Resistors

(i) Series Combination

(ii) Parallel Combination

Here,

(iii) Special Cases

If a wire of resistance ‘ R ’ is cut into ‘ n ’ equal parts and all of them are connected in parallel, then equivalent resistance become

If and be the resultant resistance of and when connected in series and parallel then

(iv) Equivalent Resistance of a Cube

(v) Resistance of a cuboidal block

Consider a cuboidal block of conducting material of dimension

(vi) Resistance of wires in the from of circles

A wire of resistance R is bent in the form of circle. The effective resistance across its diameter is

Kirchhoff’s Law In Capacitors

Junction Rule: It states that “sum of charges present on the plates of captures connected at a junction is equal to zero.

Loop Rule: It states that “in any closed loop the algebraic sum of potential drops across different elements is zero.

Charging And Discharging Of Capacitor In Series Rc Circuit

(a) Charging : When a capacitor C is connected to a battery through a resistance R , the plates of capacitor will acquire equal and opposite charge and the potential difference across it becomes equal to the unit of the battery but in opposite polarity. This process is called charging and if takes sometime and during this time there is an electric current through the resistance.

At any time t ₁ ‘ i ’ is the current following through the resistor and q is the charge on the capacitor.

But and

So, or

where

Here CR is called the capacitive time constant τ. It is the time in which charge on the capacitor reaches 0.632 times is maximum value.

Current

(b) Charging : After the completion of charging, if the battery is removed capacitor starts discharging. At any time t ,

V = IR

q = CV

At charge becomes times its initial value.

During discharging, current

where

Example 1: The figure below shows currents in a part of electric circuit. The current i is _______.

A. 1.7 amp

B. 3.7 amp

C. 1.3 amp

D. 1 amp

Sol. According to Kirchhoff’s first law

At junction A , i _AB = 2 + 2 = 4 A

At junction B , i _AB = i _BC – 1 = 3 A

At junction C , i = i _BC − 1.3 = 3 – 1.3 = 1.7 amp.

Example 2 : In the transient shown the time constant of the circuit is ________.

A. B.

C. D.

Sol.