Biot-Savart-Law - (slides).pdf

Module 19: Sources of Magnetic Fields: Biot-Savart Law

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Module 19: Outline Magnetic Fields Fields, Creating Fields: Biot-Savart Law

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Sources of Magnetic Fields

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What creates fields? Magnets – more about this later The Earth How’s How s that work? Moving charges!

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Electric Field Of Point Charge g An electric charge produces an electric field:

rˆ

rˆ

r E=

1

q ˆ r 2 4πε o r

: unit vector directed from q to P 5

Magnetic Field Of Moving Charge Moving g charge g with velocity yvp produces magnetic g field:

P

r r μo q v x rˆ B= 2 4π r ˆr : unit vector directed

rˆ

from q to P

μ0 = 4π × 10 T ⋅ m/A −7

permeability of free space 6

Animation: Field Generated by a Moving Charge

Link to animation

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Continuous C ti charge h di distributions: t ib ti ^ Currents & Biot-Savart

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From Charges to Currents?

r r dB ∝ dq v

[meter] = [coulomb] [ ] [sec] [coulomb] = [meter] [sec] r r dB ∝ Id s

r v dq

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The Biot-Savart Law Current element of length ds carrying current I produces a magnetic field:

r r μ0 I d s × rˆ dB = 2 4π r (Link to Shockwave) 10

The Right-Hand Right Hand Rule #2

ˆ ˆ zˆ × r = θ 11

Concept Questions: B fields Generated by y Currents

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Concept Question: Biot-Savart 0

The magnetic field at P points towards the

1. 2 2. 3. 4. 5 5. 6. 7 7.

+x direction +y direction +z direction -x direction -y y direction -z direction Fi ld iis zero ((so no di Field direction) ti ) 13

Concept p Question: Bent Wire The magnetic field at P is equal to the field of:

1. 2 2. 3. 4 4.

a semicircle a semicircle i i l plus l the h fifield ld off a llong straight i h wire i a semicircle minus the field of a long straight wire none off the th above b 14

Demonstration: Field Generated by y Wire

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Demonstration: D t ti Jumping Wire

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Magnetic Force on Current-Carrying Wire

Current is moving g charges, g and we know that moving charges feel a force in a magnetic field 17

Magnetic Force on Current-Carrying Wire

r r r FB = Δqv × B r Δs r Δqv = Δq Δt r Δq r = Δs = I Δs Δt r r r FB = I L × B

(

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Conceptt Question C Q ti Question: Q ti Parallel Current Carrying Wires

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Concept p Question: Parallel Wires Consider two parallel current carrying wires. With the currents running g in the same direction,, the wires are 1. 2 2. 3. 4. 5.

I

I

1

2

attracted (likes attract?) repelled ll d (likes (lik repel?) l?) pushed another direction not pushed – no net force I don don’tt know 20

Demonstration: D t ti Parallel & Anti-Parallel Anti Parallel Currents

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Can we understand why? Whether they attract or repel can be seen in the shape of the created B field

(Link to Animation)

(Link to Animation) 22

Concept Question: Current Carrying Coils

The above coils have 1. p parallel currents that attract 2. parallel currents that repel 3 opposite currents that attract 3. 4. opposite currents that repel 23

Force on Dipole from Dipole: A ti P Anti-Parallel ll l Alignment Ali t Link to animation i ti

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Force on Dipole from Dipole: Parallel Alignment Link to animation

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Example : Coil of Radius R Consider a coil with radius R and current I

I I

P I

Find the magnetic field B at the center (P)

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Example : Coil of Radius R Consider a coil with radius R and current I

I I

P I

1) Think about it: • Legs contribute nothing I parallel to r • Ring makes field into page 2) Choose a ds 3) Pick your coordinates 4) Write Biot Biot-Savart Savart

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Animation: Magnetic Field Generated by a Current Loop Link to Shockwave

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Example : Coil of Radius R In the circular part of the coil…

r r d s ⊥ ˆr → | d s × ˆr | = ds d

I I

rˆ r ds

I

Biot-Savart: Biot -Savart:

r μ0 I d s × rˆ μ0 I ds dB = = 2 4π r 4π r 2 μ0 I R dθ = 4π R 2 μ0 I dθ = 4π R 29

Example : Coil of Radius R Consider a coil with radius R and current I

I I

θ

r ds

I

μ0 I dθ dB = 4π R B = ∫ dB =

2π

∫ 0

μ0 I dθ 4π R

2π

μ0 I μ0 I = dθ = 2π ) ( ∫ 4π R 0 4π R r μ0 I B= i t page into 2R

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Example : Coil of Radius R

I I

P I

r μ0 I B= into page 2R

Notes: N t •This is an EASY Biot-Savart problem: • No N vectors t involved i l d •This is what I would expect on exam 31

Problem: B Field Fi ld from f Coil C il off Radius R di R Consider C id a coilil made d off semi-circles i i l off radii dii R and d 2R and carrying a current I What is B at point P?

P I 32

Problem: B Field Fi ld from f Coil C il off Radius R di R C Consider id a coilil with ith radius di R and d carrying i a currentt I What is B at point P? WARNING: This is much harder than the previous problem. problem Why??

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8.02SC Physics II: Electricity and Magnetism Fall 2010

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