Lecture 9-1 Lorentz Force - physics.purdue.edu
Transcript of Lecture 9-1 Lorentz Force - physics.purdue.edu
Lecture 9Lecture 9--11Lorentz Force
Let denote the electric and magnetic vector fields. The force actingon a point charge q, moving with velocity fields is:
and E B F in the superimosed v E B+
{ }F q E v B= + ×
This is called the Lorentz force equation.
Velocity Selector
Assume that the electric field is created by a parallel plate capacitor pointingalong the −Y axis and the magnetic field along the −Z axis as shown in the figure below. Since the moving charged particle is negative:
EB
(up)
(down)
Ey
By
F qE
F qvB
= +
= −
Lecture 9Lecture 9--22
When the electric and magnetic forces balance
0qE qvBEvB
− =
=
X
Y
Lecture 9Lecture 9--33
Magnetic Mass spectrometer
When an ion of unknown mass enters the homogeneous magnetic field of the Magnetic spectrometer it executes a circular path. The measured radius of the circle depends on its mass m and its velocity . Thus v also has to be measured. (See Lec.8, page 12)
BB
v
Lecture 9Lecture 9--44
Physics 219 – Question 1 – February 08, 2012.
a) Increase Eb) Increase Bc) Turn B offd) Turn E offe) Nothing
A proton (charge +e) comes horizontally into a region of perpendicularly crossed, uniform E and B fields as shown. In this region, it deflects upward as shown. What can you do to change the path so it remains horizontal through the region?
e+
Lecture 9Lecture 9--55Magnetic Force on a Current
• Consider a straight current-carrying wire in the presence of a magnetic field B.
• There will be a force on each of the charges moving in the wire. What will be the total force F on a length L of the wire?
• Current is made up of n charges/volume, each carrying charge q < 0 and moving with velocity vd through a wire of cross-section A.
qv Bו Force on each charge =
( )F n AL qv B= ו Total force:
F iL B= ×
A
• On the next page we show that the productnAvq
is equal to the current flowing in the wire.
Lecture 9Lecture 9--66
In time Δt, all the free charges in the shaded volume pass through A.If there are n charge carriers per unit volume, each with charge q, thetotal free charge in this volume is isthe drift velocity of the charge carriers.
, where d dQ qnAv t vΔ = Δ
Lecture 9Lecture 9--77Magnetic Force on a Current Loop
– Force on top path cancels force on bottom path (F = IBL)
– Force on right path cancels force on left path. (F = IBL)
loop
loop
I L B
I L B
F = ×
⎛ ⎞= ×⎜ ⎟
⎝ ⎠
∑
∑
Force on closed loop current in uniform B?
Uniform B exerts no net force on closed current loop.
closed loop
0=
Lecture 9Lecture 9--88 Magnetic Torque on a Current Loop
• If B field is ⊥ to plane of loop, the net torque on loop is 0.
r Fτ = ×definition of torque
abut a chosen point
so that n is twisted to align with B
n
n
magnetic moment direction
• If B field is // to plane of loop, the net torque on loop is maximum.
a
b
b
Lecture 9Lecture 9--99Calculation of Torque
2 sin2br F Fτ τ θ= × → = ⋅ ⋅∑
Thus: sinsin
Iab BIA B
τ θθ
= ⋅ ⋅=
• Note: if loop ⊥ B, θ = 0 and τ = 0. Maximum torque occurs when plane of the loop is parallel to B.
• Suppose the coil has width b (the side we see) and length a(into the screen). The torque about the center is given by:
μmagnetic moment
(Simple Approach)
The magnitude of magnetic force F acting on a wire of length a carrying current I in magnetic field B is: F IaB=
area of loop
Lecture 9Lecture 9--1010Calculation of Torque
(General Approach)
For reference : Giambattista, Vol. 2, Ch. 19., page 719., prob. 47.
It can be shown that the magnetic moment of planar loop of any shape of area A carrying a current I is: NIAμ =N denotes the number of turns.The magnetic torque in magnetic field is:
(19.13 )B NIA B bτ μ= × = ×The direction of is can be determined from the current direction in the loop and the right hand rule.
A
The direction of is along the rotation axis. The direction can beobtained by the right hand rule.
τ
Lecture 9Lecture 9--1111Sources of Magnetic Fields
70 2
N4 10A
πμ − ⎛ ⎞= × ⎜ ⎟⎝ ⎠
where the magnetic permeabilityconstant μ0 is
/T m A⋅also
0
2IBr
μπ
=B
r
magnetic field circulates around wire.
Since we could not find magnetic monopole, Gausstheorem for field gives:
0netB
S
B dS q⋅ = =∫However a French contemporary of Oersted,Ampere, noted that an infinite straight currentcarrying wire creates a circulating magnetic fieldaround the wire, where:
In 1820 Hans C. Oersted atUniversity of Copenhagenobserved that electric currentcreates magnetic field andlightning magnetizes iron.
0
2IBr
μπ
=
B
Lecture 9Lecture 9--1212
• Moving point charge
• (Bits of) current
I
How do we calculate B due to I?
by using calculus, or
Ampère’s Law (sometimes)
dB
In general: Biot – Savart Law
034
idl rdBr
μπ
×=
A field created by a element of a wire which conducts i current, at a distant point P from .
dldl
Lecture 9Lecture 9--1313
Circular Loop Current is a Magnetic Dipole
0( )2
IB at centerr
μ=
multiple loops ⇒
front
We have already discussed that a
Lecture 9Lecture 9--1414
Physics 219 – Question 2 – February 08, 2012.
A loop of wire lies in the x-y plane. This loop has a current I that circulates clockwise as viewed from above (from +z toward -z). What is the direction of the magnetic field at point A?
a) along +xb) along -yc) along +zd) along -ze) none of the above
z
x
y
A
I
Lecture 9Lecture 9--1515
Physics 219 – Question 3 – February 08, 2012.
A loop of wire lies in the x-y plane. This loop has a current I that circulates clockwise as viewed from above (from +z toward -z). What is the direction of the magnetic field at point A?
a) along +xb) along -yc) along +zd) along -ze) none of the above
z
x
y
A
I
Lecture 9Lecture 9--1616Solenoid’s B field synopsis
// to axis
• Long solenoid (R<<L):
B inside solenoid
B outside solenoid nearly zero
(not very close to the ends or wires)
Solenoid’s B field Bar magnet’s B field
L
R
Lecture 9Lecture 9--1717Ampere’s Law in Magnetostatics
The sum of the product of B|| (magnetic field projected along a path) and Δl (the path length) along a closed loop, Amperianloop, is proportional to the net current Inet encircled by the loop,
|| 0 netloo lp
l BB d Il μΔ = ⋅ =∫∑
• Choose a direction of summation.
• A current is positive if it flows along the RHR normal direction of the Amperian loop, as defined by the direction of summation.
0 1 2( )i iμ −
Δl
70 2
N4 10A
πμ − ⎛ ⎞= × ⎜ ⎟⎝ ⎠
Lecture 9Lecture 9--1818
// to axisB inside solenoid
B outside solenoid nearly zero
(not very close to the ends or wires)n windings per unit length
B
r
Long straight wire:
00
22 rB I IB
rπμ μπ = ∴ =
Long solenoid:
|| 0 netloop
B l IμΔ =∑
0 0( )Bh nhI B nIμ μ∴ ==
Calculation of the Solenoid Magnetic FieldWith the help of Ampere’s Law:
Lecture 9Lecture 9--1919Two Parallel Currents
12 2 2 1F I L B I LB= × =
0 11 2
IBR
μπ
=
0 1 22 2
I IF LR
μπ
∴ =
2 0 1 2 1
2F I I FL R L
μπ
= =
The ampere is defined to be the constant parallel currents 1 m apartthat will produce the force between them of 2 x 10-7 N per meter.
Definition of charge 1 C (1 A for 1 Sec)
L
1 2