Ampère's Law

Discussion

biot-savart law

This law usually no fun to deal with, but it's the elementary basis (the most primitive statement) of electromagnetism.

B =  μ0I
ds × 
r2

line of current

Bline =  µ0I
ds × 
r2
+∞
Bline =  µ0I
y/√(x2 + y2)  dx 
x2 + y2
−∞
+∞
Bline =  µ0I
y  dx 
(x2 + y2)3/2
−∞
+∞
Bline =  µ0I
x
 
y(x2 + y2)½
−∞
Bline =  µ0I
+1  −  −1
 
y y
Bline =  µ0I 2
y
Bline =  µ0I
y

loop of current (along axis)

Bloop =  µ0I
ds × 
r2
Bloop =  µ0I
a/√(x2 + a2)  a dφ 
x2 + a2
0
Bloop =  µ0I   a2
dφ 
(x2 + a2)3/2
0
Bloop =  µ0I   a2  [2π − 0] 
(x2 + a2)3/2
Bloop =  µ0I   a2  
2 (x2 + a2)3/2

Solenoid

[solenoid pic goes here]

Bsolenoid = 
dBloop
+∞
Bsolenoid =  µ0I
a2  n dx 
2 (x2 + a2)3/2
−∞
+∞
Bsolenoid =  µ0nI
x
2 √(x2 + a2)
−∞
Bsolenoid =  µ0nI  [(+1) − (1)] 
2

Bsolenoid = µ0nI 

ampère's law

Everything's better with Ampère's law (almost everything).

André-Marie Ampère (1775–1836) France

B · ds = μ0I

∇ × B = μ0J

with displacement current term

B · ds = μ0ε0  ∂ΦB  + μ0I
dt
 
∇ × B = μ0ε0  ∂ΦE  + μ0J
dt

Apply to the straight wire, infinite sheet, solenoid, toroid,… anything else?

A straight wire - look how simple it is

B · ds  =  µ0I  
 
B(2πr)  =  µ0I  
 
B  =  µ0I  
r  

A solenoid - also wonderfully simple

B · ds  =  µ0I  
 
B  =  µ0NI  
 
B  =  µ0nI  
 

Beyond the straight wire lies the infinite sheet

B · ds  =  µ0I  
 
B(2ℓ)  =  µ0σℓ
B  =  µ0σ  
2  

Beyond the solenoid lies the toroid

B · ds  =  µ0I  
 
B(2πr)  =  µ0NI  
B  =  µ0NI  
r  

Inside a wire with total current I.

B · ds  =  µ0I  
 
B(2πr)  =  µ0I πr2
πR2
B  =  µ0Ir  
R2  

Inside a wire with charge density ρ.

B · ds = µ0I = µ0
ρ dV