gauss’s law (ii) - physics & astronomy | mcmaster ...luke/1e03/lecture06.pdf · 1 gauss’s...
TRANSCRIPT
1
Gauss’s Law (II) • Examples: charged spherical shell, infinite plane, long straight wire
Text sections 24.3
Practice problems: chapter 24, problems 19, 21, 23, 27, 29, 51
Read section 24.3 carefully, including field of a line or cylinder.
For a closed surface S:
Net outward flux through S
2
1) Use symmetry to sketch the behaviour of E (possible only for very symmetic distributions)
2) Pick an imaginary “gaussian surface” S
5) Calculate the flux through S, in terms of an unknown |E|
6) Calculate the charge enclosed by S from geometry
7) Relate 3) and 4) by Gauss’s Law, solve for |E|
Total charge Q
a
r + +
+
+
+ +
+
+
+
+ +
+
+
+ +
Find E for: i) r > a and ii) r < a
E
3
P
a
r + +
+
+
+ +
+
+
+
+ +
+
+
+ +
E(r)
E(r)
E(r)
E(r)
i) r > a
P
a
r + +
+
+
+ +
+
+
+ +
+ + +
i) r > a
+ +
4
P a r +
+
+
+
+ +
+
+
+ +
+ + +
ii) r < a
P a r +
+
+
+
+ +
+
+
+ +
+ + +
ii) For r < a
Quiz For r < a, the charge enclosed within the dashed green gaussian sphere is proportional to
A) r B) r2
C) r3
D) is independent of r
5
P a r +
+
+
+
+ +
+
+
+ +
+ + +
ii) r < a
For any spherically – symmetric charge:
Where Q(r ) = charge enclosed within an (imaginary) sphere of radius r
6
+ + +
+
+
+
+ + +
+ +
+
+
r1
r2
a
P1
P2
Total charge Q
Outside: (r1 > a):
Inside: (r2 < a):
+ + +
+
+
+
+ + +
+ +
+
+ r2
a
P2
Inside the shell, we expect the field to be:
A) zero everywhere inside B) zero only at the centre, and
pointing outwards from the centre to the wall
C) zero at the centre, and pointing inwards from the wall to the centre
7
+ + +
+
+
+
+ + +
+ +
+
+
r1
r2
a
P1
P2
Total charge Q
Outside: (r1 > a):
Inside: (r2 < a):
+ + + + + + + +
+ + +
E
gaussian surface
8
+ + + + + + + +
+ + +
E
gaussian surface 1) -Same on both sides.
2) Gaussian surface: cylinder, faces||sheet
+ + + + area A
area A E
+ + + + + + + +
+ + +
1) -Same on both sides.
2) Gaussian surface: cylinder, faces||sheet
+ + + + area A
area A E
9
E (1)
(2)
(3)
(3) Flux:
(4) Qenclosed =
Φ1 = Φ2 = Φ3 =
ΦTOTAL =
A1
A2
E (1)
(2)
(3)
(3) Flux:
(4) Qenclosed
ΦTOTAL = 2EA1
A1
A2
Φ1 = EA1 (E ⊥ surface) Φ2 = EA2 = EA1 Φ3 = 0 (E || side)
10
Using Gauss’s Law…
Infinite sheet of charge
+ + + + + +
Quiz
The electric field 10 cm above an infinite, uniformly-charged plane is 100 N/C. At a point 20 cm above the plane, the field would be
A) zero B) 100 N/C C) 50 N/C D) 25 N/C
11
+ + + + + + + + + r
L Flux:
Qenclosed:
1) Symmetry: - is radially outwards (inwards??)
- depends only on
2) Pick a “gaussian surface”: sphere of radius r
3)
( at each point on S)
( constant on S)
12
4) 5) Gauss’s Law:
When r < a (inside the ball):
P a r +
+
+
+
+ +
+
+
+ +
+ + +
• Guassian surface: sphere of radius r • Flux:
• Charge enclosed: