EQUATION OF CONTINUITY AND KIRCHHOFF'S CURRENT LAW

EQUATION OF CONTINUITY AND KIRCHHOFF'S CURRENT LAW

1. D.Gopinath AP/ECE
Ramco Institute of technology
Academic year (2018-19 Even)

2. Due to the principle of charge conservation, the time
rate of decrease of charge within a given volume must
be equal to the net outward current flow through the
closed surface of the volume.

Thus current I
out
coming out of the closed surface is

3. where Qin is the total charge enclosed by the closed
surface. Invoking the divergence theorem

4. Comparing the above equations,
which is called the continuity of current equation.

5. For steady currents,


And hence

showing that the total charge leaving a volume is the
same as the total charge entering it. Kirchhoff's current
law follows from this.

6. To consider the effect of introducing charge at some
interior point of a given material (conductor or
dielectric),
Invoking Ohm’s Law


and Gauss Law,

7. By using the above two laws into Continuity equation,
we get

8. The above equation is a homogeneous linear ordinary
differential equation. By separating variables, we get



And integrating both sides gives,

9. Taking Exponential on both sides

ρ
v0
is the initial charge density (i.e., ρ
v at t = 0).

10. The above equation shows that as a result of
introducing charge at some interior point of the
material there is a decay of volume charge density ρ
v.
The time constant T
r (in seconds) is known as the
Relaxation time or Rearrangement time.

Relaxation time is the time it takes a charge placed in
the interior of a material to drop to e
-1
=36.8 percent of
its initial value.

12. TEXT BOOKS:
1. William H Hayt and Jr John A Buck, “Engineering
Electromagnetics” , Tata McGraw-Hill Publishing Company
Ltd, New Delhi, 2008
2. Sadiku MH, “Principles of Electromagnetics”, Oxford
University Press Inc, New Delhi, 2009

REFERENCES:

1. David K Cheng, “Field and Wave Electromagnetics”,
Pearson Education Inc, Delhi, 2004
2. John D Kraus and Daniel A Fleisch, “Electromagnetics
with Applications”, McGraw Hill Book Co,
3. Karl E Longman and Sava V Savov, “Fundamentals of
Electromagnetics”, Prentice Hall of India, New Delhi, 2006
4. Ashutosh Pramanic, “Electromagnetism”, Prentice Hall of
India , New Delhi, 2006