Introduction to physics Lecture29_2024.pdf

Shiv Nadar University, 2nd semester course.
Introduction to physics 2
Lectures notes 29

1. Lecture 29
Introduction to magnetic
vector potential

3. Useful Comparison: E and B fields
Electric field line has a definite starting point and a definite termination point.
But lines representing a field of induction B have different properties- they
always form closed loops. We cannot say where a line of field B starts or where
it ends.
Through any closed surface S, the total
incoming flux is exactly equal to the total
outgoing flux
Φ=ර
??????
�∙????????????=0

4. ??????×??????=0 permits us to introduce a scalar potential ?????? in electrostatics
(Curl of any divergence is zero ??????×????????????=0 ).
⇒??????=−????????????
Similarly, ??????∙�=0 invites the introduction of vector potential � in
magnetostatics (Divergence of any curl is zero ??????∙??????×�=0 ).
⇒�=??????×�
Just as you can add to V, any function whose gradient is zero (i.e constant)
without altering E, similarly you can add to A, any function whose curl
vanishes (i.e gradient of any scalar) with no effect on B.
Choose �=�
??????+??????� � is any scalar
Ampere’s law:
(1)
(2)
Vector Potential

5. We can always choose � in such a way so that
This is similar to Poisson equation ⇒
??????∙�=0
(2)
from (1)
Line currents Surface currents
⇒ ⇒
⇒

6. Direction of A is the direction of current
Here we can use another trick to avoid integration up to infinite.

8. Magnetostatic Boundary Conditions
Summary of Magnetostatics
Isn’t it similar to electrostatics? Are B and A continuous over
a current carrying surface?

9. Just as the electric field suffers discontinuity at a surface charge, similarly
magnetic fields and potentials suffers discontinuity at surface currents
ර�.??????�=0??????
⊥
����??????
=??????
⊥
�????????????��
ර�.????????????=�
0??????
??????��
??????
∥
����??????
−??????
∥
�????????????��
=�
0??????
Normal components of B are continuous
at surface of current density K
Tangential components of B are
discontinuous at surface currents K
ෝ??????
Combining

10. Magnetostatic Boundary Conditions: Detailed
Calculations (Optional)

11. Magnetostatic Boundary Conditions: Detailed
Calculations

12. Magnetostatic Boundary Conditions: Detailed
Calculations

13. Like the scalar potential in electrostatics, vector potential is continuous
across any boundary
But derivative of A inherits the discontinuity of B
Can be proved from the
relations ??????.�=0 and �=??????�