# El campo magnético de una corriente

## Texto completo

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

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

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2

2

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2

3

### ˆ

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µ0: permeabilidad del vacío 2 7 7 0 4 10 Tm/ A 4 10 N / A − − = = π π µ

3 0

2 0

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### por una carga en movimiento

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Campo magnético creado por corrientes eléctricas: Ley de Biot y Savart

2 0

2 0

3 0

………. 2 0

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### Sources of Electric field, magnetic field

From Coulomb's Law, a point charge dq

generates electric field distance away from the source:

2 0

### r

r r r dE dq P

From Biot-Savart's Law, a point current

generates magnetic field distance away from the source:

r dB

0 2

### r

Difference:

1. A current segment , not a point charge dq, hence a vector.

2. Cross product of two vectors, is determined by the right-hand rule, not .

r

Ids

r dB

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## ∫

### µ

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2 0 ˆ 4 r Id dB = l×r π µ φ π µ sen r Idx dB 0 2 4 = Líneas de campo magnético creadas por

un hilo conductor θ π µ cos 4 2 0 r Idx dB = θ ytg x= dx = ysecdθ θ cos r y = θ θ π µ d y I dB cos 4 0 = ( 1 2) 0 4π θ θ µ sen sen y I B = +

Para un conductor muy largo:

2 2 1 π θ θ = = y I B π µ 2 0 =

) (π φ

φ = sen

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### magnetic field goes in circles

 The magnetic field lines are

circles concentric with the wire

 The field lines lie in planes

perpendicular to to wire

 The magnitude of the field is

constant on any circle of radius a

 A different and more

convenient right-hand rule for determining the direction of the field is shown

### B

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Campo magnético creado por una espira circular 2 0 ˆ 4 r Id dB = l×r π µ 2 2 0 2 0 4 ˆ 4 x R Idl r Id d + = × = π µ π µ l r B 2 2 2 2 0 2 2 4 R x R R x Idl R x R dB dBsen dBx + + =         + = = π µ θ

### ∫

+ = + = = dl R x IR dl R x IR dB Bx x 3/2 2 2 0 2 / 3 2 2 0 4 4 π µ π µ

2 2

### )

3/2 0 (2 ) 4 x R R IR Bx + =

### µ

Líneas de campo creado por una espira circular Líneas de campo creado por una espira circular

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

I I I = 2

### ∫

= 2 2 = 4 4 2 o o o full circle µ µ µ B ds πr πr πr r

### π

= × = r r 0 0 2 2 4 4 Id Ids ˆ ˆ d r r s B r k

### Ids

r Ids O Y X r r

Off center points are not so easy to calculate.

o



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### Example problems

A long, straight wire carries current I. A right-angle bend is made in the middle of the wire. The bend forms an arc of a circle of radius r. Determine the magnetic filed at the center of the arc.

Formula to use: Biot-Savart’s Law, or more specifically the results from the discussed two examples: I I sin = 2 = 0 4 4 π o o µ µ B θ πr πr

For the straight section

For the arc = I

### ∫

= I = I

1 4 2 2 2 4 4 4 8 o o o circle µ µ πr µ B ds πr πr r

The final answer: = I + I + I = I  + 

  2 1 4 8 4 4 2 o o o o µ µ µ µ B πr r πr r π magnitude

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2 0 ˆ 4 r Id dB = l×r π µ φ π µ sen r Idx dB 0 2 4 = Líneas de campo magnético creadas por

un hilo conductor θ π µ cos 4 2 0 r Idx dB = θ ytg x= dx = ysecdθ θ cos r y = θ θ π µ d y I dB cos 4 0 = ( 1 2) 0 4π θ θ µ sen sen y I B = +

Para un conductor muy largo:

2 2 1 π θ θ = = y I B π µ 2 0 =

) (π φ

φ = sen

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### Conductors

 Two parallel wires each carry

 The field due to the current in

wire 2 exerts a force on wire 1 of

F1 = I1B

2

 Substituting the equation for B2

gives



### rule:

 same direction currents attract

each other

 opposite directions currents

repel each other

2 B r I I a = 1 2 l 1 2 o µ F π

The force per unit length on the wire is I I

a = l 1 2 2 o B µ F π

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### Force on a Wire, the formula

 The magnetic force is exerted on each

moving charge in the wire



 The total force is the product of the force

on one charge and the number of charges in the wire

 d q = × F v B r r r

### )

= × = × r r r r r d d q nAL qnA L F v B v B

 In terms of the current, this becomes

 I is the current

 is a vector that points in the direction of the

current

 Its magnitude is the length L of the segment

B

### r

L r d I = nq ⋅ r r Q v A

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Campo magnético creado por un solenoide

2 2

### )

3/2 2 0 2 / 3 2 2 2 0 2 4 ) 2 ( 4 x R nIdx R R x R di dBx + = + = π π µ π π µ Líneas de campo debidas a dos espiras que transportan la misma corriente en el mismo sentido

### ∫

+ = b a x R x dx nI R B 3/2 2 2 2 0 2 4π π µ       + + + = 2 2 2 2 0 2 1 R a a R b b nI Bx µ nIdx di = L N n = /

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Campo magnético creado por una espira circular 2 0 ˆ 4 r Id dB = l×r π µ 2 2 0 2 0 4 ˆ 4 x R Idl r Id d + = × = π µ π µ l r B 2 2 2 2 0 2 2 4 R x R R x Idl R x R dB dBsen dBx + + =         + = = π µ θ

### ∫

+ = + = = dl R x IR dl R x IR dB Bx x 3/2 2 2 0 2 / 3 2 2 0 4 4 π µ π µ

2 2

### )

3/2 0 (2 ) 4 x R R IR Bx + =

### µ

Líneas de campo creado por una espira circular Líneas de campo creado por una espira circular

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### r

Choose the Gauss’s Surface, oops, not again!

Choose the Ampere’s loop, as a circle with radius a.

Ampere’s Law says

o

o o

### π

is parallel with , so r B r ds

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



### ampere’s circle

2 2 o o µ d B πr µ B πr ⋅ = = → =

### ∫

B s r r I ( ) I 2 2 2 2 2 o o r d B πr µ R µ B r πR ⋅ = = → =   =  

### ∫

B s r r ( ) I ' I ' I I

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





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

### approached when:

 the turns are closely spaced

 the length is much greater than



⋅ = ⋅ = =

r r

r r

### ∫

l 1 1 path path d d B ds B B s B s o d B

NI ⋅ = =

B s r r l

o o

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





o o

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## Referencias

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