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III–2 Magnetic Fields Due to Currents

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Main Topics

• Forces on Moving Electric Charges

• Biot-Savart Law

• Ampere’s Law.

• Calculation of Some Magnetic Fields.

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Forces on Moving Electric Charges I

• Since currents are in reality moving charges it can be expected that all what is valid for interaction of magnetic fields with currents will be valid also for moving charges.

• The force of a magnetic field acting on a charge q moving by a velocity is given by the Lorentz formula:

)( BvqF

B

vF

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Forces on Moving Electric Charges II

• Lorentz force is in fact part of a more general formula which includes both electric and magnetic forces:

• This relation can be taken as a definition of electric and magnetic forces and can serve as a starting point to study them.

)]([ BvEqF

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Forces on Moving Electric Charges III

• Lorentz force is a central issue in whole electromagnetism. We shall return to it by showing several examples. Moreover we shall find out that it can be used as a basis of explanation of almost all magnetic and electromagnetic effects.

• But at this point we need to know how are magnetic fields created quantitatively.

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Biot-Savart Law I

• There are many analogies between electrostatic and magnetic fields and of course a question arises whether some analog of the Coulomb’s law exists, which would describe how two short pieces of wires with current would affect themselves. It exists but it is too complicated to use. For this reason the generation and influence of magnetic fields are separated.

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Biot-Savart Law II

• All what is necessary to find the mutual forces of two macroscopic wires of various sizes and shapes with currents is to employ the principle of superposition, which is valid in magnetic fields as well and integrate.

• It is a good exercise to try to make a few calculations then try do something better!

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Magnetic Field Due to a Straight Wire I

• Let’s have an infinite wire which we coincide with the x-axis. The current I flows in the +x direction. We are interested in magnetic induction in the point P [0, a].

• The main idea is to use the principle of superposition. Cut the wire into pieces of the same length dx and add contribution of each of them.

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Magnetic Field Due to a Straight Wire II

• For a contribution from a single piece we use formula derived from the Biot-Savart law:

• Since both vectors which are multiplied lie in the x, y plane only the z component of will be non-zero which leads to a great simplification. We see where the right hand rule comes from!

30

4xP

xP

r

rxdIBd

B

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Magnetic Field Due to a Straight Wire III

• So a piece of the length dx with the coordinate x contributes:

• Here r is the distance of dx and P and is the angle between the line joining dx and P and the x-axis. We have to express all these quantities as a function of one variable e.g. the .

20 sin

4 r

dxIdBz

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Magnetic Field Due to a Straight Wire IV

• For r we get:

and for x and dx (- is important to get negative x at angles < /2 !):

2

2

2

sin1sin

arar

2sincotcot

dadxax

a

x

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Magnetic Field Due to a Straight Wire V

• So finally we get:

The conclusions we can derive from the symmetry we postpone for later!

a

Id

a

I

a

daIBz

2sin

4

sin

sinsin

4

0

0

0

022

20

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Ampère’s Law

• As in electrostatics also in magnetism a law exists which can considerably simplify calculations in cases of a special symmetry and can be used to clarify physical ideas in many important situations.

• It is the Ampères law which relates the line integral of over a closed path with currents which are surrounded by the path.

B

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Magnetic Field Due to a Straight Wire VI

• As it is the case with using the Gauss’ law, we have to find a path which is tangential to everywhere and on which the magnitude of B is constant. So it must be a special field line. Then we can move B out of the integral, which then simply gives the length of the particular integration path.

B

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Magnetic Field Due to a Straight Wire VII

• Let us have a long straight wire with current I.

• We expect B to depend on r and have axial symmetry where the wire is naturally the axis.

• The field lines, as we already know are circles and therefore our integration path will be a circle with a radius r equal to the distance where we want to find the field. Then:

r

IrB

IrrB

2)(

)(2

0

0

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Magnetic Field Due to a Straight Wire VIII

• The vectors of the magnetic induction are tangents to circles centered on the wire, which thereby are the field lines, and the magnitude of B decreases with the first power of the distance. • It is similar as in the case of the electrostatic

field of an straight, infinite and uniformly charged wire but there electric field lines were radial while here magnetic are circular, thereby perpendicular in every point.

B

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Magnetic Field in a Center of a Square Loop of Current I

• Apparently by employing the Amperes law we have obtained the same information in a considerable easier way. But, unfortunately, this works only in special cases.

• Let’s calculate magnetic induction in the center of a square loop a x a of current I. We see that it is a superposition of contributions of all 4 sides of the square but to get these we have to use the formula for infinite wire with appropriate limits.

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Magnetic Field in a Center of a Square Loop of Current II

• The contribution of one side is:

etc.

22

sin4

0

2

04

3

4a

Id

IB

az

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Force Between Two Straight Wires I

• Let us have two straight parallel wires in which currents I1 and I2 flow in the same direction separated by a distance d.

• First, we can find the directions and then simply deal only with the magnitudes. It is convenient to calculate a force per unit length.

d

II

l

F 210

2

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Force Between Two Straight Wires II

• This is used for the definition of 1 ampere:

1 ampere is a constant current which, if maintained in two straight parallel conductors of infinite length, of negligible cross section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 10-7 N per meter of length.

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Homework

• No homework!

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Things to read

• This Lecture Covers

Chapter 28 – 1, 2, 3, 4, 6

• Advance reading

Chapter 27 – 5; 28 – 4, 5

Magnetic interaction of two currents I

312

1212210212 ||4

)]([)(

rr

rrldldIIrFd

Let us have two currents I1 and I2 flowing in two short straight pieces of wire and Then the force acting on the second piece due to the existence of the first piece is:

This very general formula covers almost all the magnetism physics but would be hard to use in practice.

)( 11 rld

)( 22 rld

Magnetic interaction of two currents II

BdldIrFd

22212 )(

That is the reason why it is divided into the formula using the field (we already know):

and the formula to calculate the field, which particularly is the Biot-Savart law:

312

121102 ||4

)]([)(

rr

rrldIrBd

Magnetic interaction of two currents III

If we realize that:

is a unit vector pointing in the direction from the first current to the second one , we se that magnetic forces decrease also with the second power of the distance.

212

012110

2 ||4

][)(

rr

rldIrBd

||

)(

12

12012 rr

rrr

1r

2r

Magnetic interaction of two currents IV

The “scaling” constant 0 = 4 10-7 Tm/A is called the permeability of vacuum or of free space. Some authors don’t use it since it is not an independent parameter of the Nature. It is related to the permitivity of vacuum 0 and the speed of light c by:

200

1

c

^

Ampère’s Law

iIldB 0

Let us have none, one, two ore more wires with currents I1, I2 … then:

• All the current must be added but their polarities must be taken into account !

^