Chapter 3.2: Ampere’s Circuital Law and its Applications
Discipline Course-I
Semester-II
Paper No: Electricity and Magnetism
Lesson: Chapter 3.2: Ampere’s Circuital Law and its
Applications
Lesson Developer: Dr Namrata Soni
College/ Department: Hans Raj College, University of Delhi
Institute of Lifelong Learning, University of Delhi
,LEARNING OBJECTIIVES
After going through this chapter, the reader would be able to
Understand Ampere’s circuital law and its importance in calculating the magnetic
field.
Appreciate the limitations of Ampere’s circuital law.
Use Ampere’s circuital law for calculating the magnetic field due to a current
carrying solenoid and a toroid.
Do a direct calculation of div and understand the significance of the result.
Do a direct calculation of curl and understand how this result leads us to Ampere’s
circuital law.
Get familiar with the concept of vector potential of magnetic field.
Understand the significance and importance of the concept of ‘vector potential’ in
calculation of the magnetic field due to a given current distribution.
, INTRODUCTION
We start this chapter by stating the Ampere’s circuital law. The relevance of this law for
calculating magnetic fields, due to symmetric current distributions, is explained by taking
the examples of a solenoid and a toroid. The fact that this law can be regarded as an
alternative way of the Biot Savart’s law, is brought out through an exclusive calculation of
the curl of . This calculation, incidentally also bring out the inherent limitation of the usual
form of this law i.e. its validity holding only for steady currents. We also do a direct
calculation of the divergence of and find out that .The physical significance of this
result i.e. the non-existence of the isolated magnetic poles, is also discussed and explained.
A brief discussion on the concept of ‘vector potential’, , for the magnetic fields, helps
us to appreciate the role of this concept in maintaining a ‘superficial symmetry’ between
electric and magnetic fields. The relation between and is brought out and the use of
in simplifying the calculations of magnetic fields, for any general current distribution, is
briefly discussed.
Ampere’s circuital law
This law due to Ampere, provides us with an alternative way of calculating the
magnetic field due to a given current distribution. This law, is, in a way, similar to the
Gauss’s law in electrostatics, which again provides us with an alternative way of calculating
the electric field due to a given charge distribution.
Ampere’s circuital law states: The line integral of the magnetic field, over a closed
path, or loop, equals times the total current enclosed by that closed loop. We express this
law through the mathematical expression:
where , is the net current enclosed by the loop ‘ ’.
We shall be discussing the details of a ‘proof’ of this law (through our calculation of
curl of the magnetic field, ), later on in this unit. We shall than realize that this law can be
regarded as an alternative way of stating the Biot Savart’s law. We shall also see that the
law stated in its above form, is valid only for steady currents. The subsequent generalization
of this law, for non-steady currents, by Maxwell, played a crucial role in the development of
the electromagnetic theory of light.
A ‘Built-in Limitation’ of Ampere’s circuital law
Ampere’s law, as stated above, provides us with an easy, quick and convenient way of
calculating the line integral of the magnetic field over a given closed path or loop. We only
need to know, or calculate, the (net) current enclosed by that loop. However we need to
know the magnetic field itself, rather than just a value of its ‘line integral’. Therefore, we
need to a special closed path for which one can calculate the magnetic field, rather than just
its line integral, through the circuital law. The choice of such a (special) closed path
becomes possible only for a limited range of current distributions that have some sort of
symmetry or ‘idealization’ associated with them. Ampere’s circuital law, therefore, is a
handy tool for calculating the magnetic field only for a (very much) limited range of current
distributions. This ‘built in’ limitation, of this law, restricts its use in practical situation.
Discipline Course-I
Semester-II
Paper No: Electricity and Magnetism
Lesson: Chapter 3.2: Ampere’s Circuital Law and its
Applications
Lesson Developer: Dr Namrata Soni
College/ Department: Hans Raj College, University of Delhi
Institute of Lifelong Learning, University of Delhi
,LEARNING OBJECTIIVES
After going through this chapter, the reader would be able to
Understand Ampere’s circuital law and its importance in calculating the magnetic
field.
Appreciate the limitations of Ampere’s circuital law.
Use Ampere’s circuital law for calculating the magnetic field due to a current
carrying solenoid and a toroid.
Do a direct calculation of div and understand the significance of the result.
Do a direct calculation of curl and understand how this result leads us to Ampere’s
circuital law.
Get familiar with the concept of vector potential of magnetic field.
Understand the significance and importance of the concept of ‘vector potential’ in
calculation of the magnetic field due to a given current distribution.
, INTRODUCTION
We start this chapter by stating the Ampere’s circuital law. The relevance of this law for
calculating magnetic fields, due to symmetric current distributions, is explained by taking
the examples of a solenoid and a toroid. The fact that this law can be regarded as an
alternative way of the Biot Savart’s law, is brought out through an exclusive calculation of
the curl of . This calculation, incidentally also bring out the inherent limitation of the usual
form of this law i.e. its validity holding only for steady currents. We also do a direct
calculation of the divergence of and find out that .The physical significance of this
result i.e. the non-existence of the isolated magnetic poles, is also discussed and explained.
A brief discussion on the concept of ‘vector potential’, , for the magnetic fields, helps
us to appreciate the role of this concept in maintaining a ‘superficial symmetry’ between
electric and magnetic fields. The relation between and is brought out and the use of
in simplifying the calculations of magnetic fields, for any general current distribution, is
briefly discussed.
Ampere’s circuital law
This law due to Ampere, provides us with an alternative way of calculating the
magnetic field due to a given current distribution. This law, is, in a way, similar to the
Gauss’s law in electrostatics, which again provides us with an alternative way of calculating
the electric field due to a given charge distribution.
Ampere’s circuital law states: The line integral of the magnetic field, over a closed
path, or loop, equals times the total current enclosed by that closed loop. We express this
law through the mathematical expression:
where , is the net current enclosed by the loop ‘ ’.
We shall be discussing the details of a ‘proof’ of this law (through our calculation of
curl of the magnetic field, ), later on in this unit. We shall than realize that this law can be
regarded as an alternative way of stating the Biot Savart’s law. We shall also see that the
law stated in its above form, is valid only for steady currents. The subsequent generalization
of this law, for non-steady currents, by Maxwell, played a crucial role in the development of
the electromagnetic theory of light.
A ‘Built-in Limitation’ of Ampere’s circuital law
Ampere’s law, as stated above, provides us with an easy, quick and convenient way of
calculating the line integral of the magnetic field over a given closed path or loop. We only
need to know, or calculate, the (net) current enclosed by that loop. However we need to
know the magnetic field itself, rather than just a value of its ‘line integral’. Therefore, we
need to a special closed path for which one can calculate the magnetic field, rather than just
its line integral, through the circuital law. The choice of such a (special) closed path
becomes possible only for a limited range of current distributions that have some sort of
symmetry or ‘idealization’ associated with them. Ampere’s circuital law, therefore, is a
handy tool for calculating the magnetic field only for a (very much) limited range of current
distributions. This ‘built in’ limitation, of this law, restricts its use in practical situation.
