Download Biot-Savart law

GANDHINAGAR INSTITUTE OF TECHNOLOGY
Electromagnetic
Active Learning Assignment
130120111010 Mihir Hathi
130120111012 Komal Kumari
Guided By: Prof. Mayank Kapadiya
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Branch : Electronic and Communication
Engineering Div: E
CONTENT

1.
2.
3.
4.
5.
6.
7.
Biot-Savart Law
Introduction
Set-Up
Observations
Equation
Total Magnetic Field
Final Notes
Special Cases
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



Magnetic Flux
Gauss’ Law in Magnetism
Ampere’s Law – General Form
STOKES’ THEOREM
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BIOT-SAVART
LAW
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INTRODUCTION
Biot and Savart conducted experiments on the force exerted
by an electric current on a nearby magnet
 They arrived at a mathematical expression that gives the
magnetic field at some point in space due to a current

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SET-UP
The magnetic field is dB at
some point P
 The length element is ds


Please replace with
fig. 30.1
The wire is carrying a steady
current of I
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OBSERVATIONS
The vector dBis perpendicular to both dsand to the unit
vector r̂ directed from dB toward P
 The magnitude of ds is inversely proportional to r2, where r
is the distance from dsto P
 The magnitude of dBis proportional to the current and to
the magnitude ds of the length element ds
 The magnitude of dBis proportional to sin q, where q is
the angle between the vectors r̂ and ds

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EQUATION
The observations are summarized in the mathematical
equation called the Biot-Savart law:
μo I ds  ˆr
dB 
4π
r2
 The magnetic field described by the law is the field due to
the current-carrying conductor


Don’t confuse this field with a field external to the conductor
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PERMEABILITY OF FREE SPACE

The constant mo is called the permeability of free space
mo = 4p x 10-7 T. m / A
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TOTAL MAGNETIC FIELD

dBis the field created by the current in the length
segment ds
 To find the total field, sum up the contributions from all
the current elements I ds
μo I
ds  ˆr
B
4π  r 2

The integral is over the entire current distribution
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FINAL NOTES
The law is also valid for a current consisting of charges
flowing through space
 dsrepresents the length of a small segment of space in
which the charges flow


For example, this could apply to the electron beam in a TV set
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BFOR A LONG, STRAIGHT CONDUCTOR
 The
thin, straight wire is
carrying a constant current

ds ˆr   dx sin θ  k̂
 Integrating
over all the
current elements gives
μo I θ2
B
cos θ dθ

θ
4πa 1
μo I

 sin θ1  sin θ2 
4πa
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B FOR A LONG, STRAIGHT CONDUCTOR, SPECIAL
CASE
 If
the conductor is an
infinitely long, straight
wire, q1 = p/2 and
q2 = -p/2
 The field becomes
μo I
B
2πa
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BFOR A CURVED WIRE SEGMENT
 Find
the field at point O
due to the wire segment
 I and R are constants
μo I
B
θ
4πR

q will be in radians
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SPECIAL CASES
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BFOR A CIRCULAR LOOP OF WIRE

Consider the previous result, with a full circle


q = 2p
μo I
μo I
μo I
B
θ 
2π 
4πa
4πa
2a
This is the field at the center of the loop
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BFOR A CIRCULAR CURRENT LOOP
 The
loop has a radius of R
and carries a steady
current of I
 Find the field
μo I at
a 2point P
Bx 

2 a x
2
2

3
2
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MAGNETIC FLUX
 The
magnetic flux
associated with a magnetic
field is defined in a way
similar to electric flux
 Consider an area element
dA on an arbitrarily
shaped surface
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The magnetic field in this element is B
 dA is a vector that is perpendicular to the surface
 dA has a magnitude equal to the area dA
 The magnetic flux FB is

B 

 B  dA
The unit of magnetic flux is T.m2 = Wb

Wb is a weber
In which the magnetic flux density (or magnetic induction) in free
space is:
and where the free space permeability is
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GAUSS’ LAW IN MAGNETISM

Magnetic fields do not begin or end at any point


The number of lines entering a surface equals the number of
lines leaving the surface
Gauss’ law in magnetism says the magnetic flux
through any closed surface is always zero:


B

d
A

0

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AMPERE’S CIRCUITAL LAW
In classical electromagnetism, Ampère's circuital law, discovered
by André-Marie Ampère in 1826,relates the integrated magnetic
field around a closed loop to the electric current passing through the
loop. James Clerk Maxwell derived it again using hydrodynamics in
his 1861 paper On Physical Lines of Force and it is now one of
the Maxwell equations, which form the basis of
classical electromagnetism.
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Where J is the total current density (in ampere per
square metre, Am−2).
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where Jf is the free current density only. Furthermore
• is the closed line integral around the closed curve C,
• denotes a 2d surface integral over S enclosed by C
•dℓ is an infinitesimal element of the curve C (i.e. a vector with magnitude
equal to the length of the infinitesimal line element, and direction given by
the tangent to the curve C)
•dS is the vector area of an infinitesimal element of surface S
STOKES’ THEOREM
The theorem is named after the Irish mathematical
physicist Sir George Stokes (1819–1903).
 What we call Stokes’ Theorem was actually discovered by
the Scottish physicist Sir William Thomson (1824–1907,
known as Lord Kelvin).
 Stokes learned of it in a letter from Thomson in 1850.

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LET
 S be an oriented piecewise-smooth surface bounded by a simple,
closed, piecewise-smooth boundary curve C with positive orientation
 F be a vector field whose components have continuous partial
derivatives on an open region in R3 that contains S.
 Then,
∫
F
⋅
dr
=
C
S

∫∫
curl F ⋅dS
Thus, Stokes’ Theorem says:
The line integral around the boundary curve of S of the tangential
component of F is equal to the surface integral of the normal
component of the curl of F.
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