The Biot-Savart law

The Biot-Savart law serves to calculate magnetic field for situations where Ampere's law is not useful due to a lack of symmetry. If the Coulomb law for the electric field of a point charge represents the underlying basis for Gauss' law of electrostatics, the Biot-Savart law plays the same role with respect to Ampere's law.

One can ask the question how the macroscopic magnetic field generated by a current passing through a wire is composed of infinitesimal contributions d B generated by oriented infinitesimal line segments d l . One can motivate the Biot-Savart law by performing the direct calculation for a straight-line wire, and then realize that the generalization for the magnetic field contribution at displacement vector r from the line segment d l reads as

d B = mu[0]/4 Pi I[0]*crossprod(d l , r )/r^3

Note the 1/ r ^2 dependence of this expression, which upon integration leads to a 1/ r dependence of the field strength B . This is analogous to the electric field generated by a charged rod. Note, however that the direction of the E field is radial, while the B field is orthogonal to the line segment d l as well as to the displacement vector r (as evident from the cross product).

Our interest is to make practical use of the Biot-Savart law. We calculate the magnetic field due to a current passing through a finite straight wire of length L at a distance D away from the middle.

We orient the wire along the x axis through the origin with - L /2 < x < L /2. Consider the triangle formed from the displacement x (location of d l =[dx, 0, 0]), the point where the field is to be calculated ([0, D , 0]) and the origin. The separation is given as r=sqrt(x^2+D^2) , the sine of the angle between r and d l required for the magnitude of the cross product can be expressed as sin(theta)=D/r .

A bit of geometry shows that the following integral is to be calculated:

> r:=sqrt(x^2+D^2);

> B:=mu[0]*I[0]/(4*Pi)*Int(D/r*(1/r^2),x=-L/2..L/2);

> B0:=value(B);

In the limit of an infinitely long wire the result previously calculated from Ampere's law is obtained:

> Blim:=limit(B0,L=infinity);

Exercise:

Use quantitative numerical examples to investigate the difference beween both expressions: calculate the field strength at distances away from the wire that are fractions (multiples) of the wire length. How different are the results?

Exercise:

Consider a current-carrying circular loop with radius R . Calculate the magnetic field at the center of the loop. Insert numerical values for the radius (in the range of 0.1 m) and the current (in the several A range). How does your result compare to the Earth's magnetic field at the surface (about 0.04 mT)?

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The current loop: calculation of Bz(0,z)

We consider now the following problem: a charge-carrying circular loop with radius R produces a magnetic field perpendicular to the plane of the loop. Calculate the strength of the magnetic field at a distance d away over the center of the loop.

We place the loop in the x - y plane with the center at the origin. An analysis of the Biot-Savart law shows that along the z -axis the following is true: while there are transverse component contributions in d B from individual segments d l (pointing in the x - y plane), their sum vanishes due to symmetry (they all cancel each other). This is apparent from the cross product between r and d l , and the rotational symmetry of the problem. Thus, we need to integrate only the z -projection of the induced field. The required direction cosine can be expressed as R / r . The integration over the polar angle phi to complete the circular path is trivial due to rotational symmetry. The vectors r and d l are orthogonal, i.e., the magnitude of the cross product involves no angle. These arguments are illustrated with diagrams in first-year physics texts. In the worksheet Currentloop.mws we carry out the line integral calculation for the case of an arbitrary location ( x , y , z ).

We have:

> r:=sqrt(R^2+d^2);

> B:=mu[0]*I[0]/(4*Pi)*2*Pi*R*(1/r^2)*R/r;

At distances d >> R the numerator of this expression can be simplified to

> Blim:=subs(R^2+d^2=d^2,B);

> interface(showassumed=0): assume(d>0):

> Blim:=simplify(Blim);

This result is important insofar as it describes the magnetic dipole field associated with a current loop. The expression represents a useful approximation to the magnetic field for large distances d away from the source (electromagnetic current loop, or a short permanent magnet).

Permanent magnetism can be understood as the net result of many aligned microscopic spins (electron spin is a quantum phenomenon). One calls the product of current times the area of the loop that appears in the numerator the magnetic moment associated with the current loop. It is worth remembering that the strength of the magnetic field along the axis behaves as the dipole moment divided by the distance cubed (times mu[0]/(2*Pi)).

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Maxwell's displacement current

Maxwell generalized Ampere's law to include the situation when a current deposits charge (Maxwell displacement current). It is the modified Ampere law that includes this current which enters the comprehensive treatment of electromagnetic fields.

To understand the problem consider a capacitor in the process of being charged, i.e., a time-dependent current flows (until the plates are fully charged). The current flows through the wire connecting plate 1, no current flows between the plates, and a current flows again from plate 2 into its connecting wire.

It is possible to consider two surfaces bounded by the same closed contour around the wire:

surface 1 is defined by the circle, with the wire carrying current I[0] passing through.

Let surface 2 be the surface surrounding the capacitor plate (imagine an inflated balloon with the capacitor plate inside, the (missing) open bottom being surface 1).

Clearly no current passes through surface 2, yet it is bounded by the same circular path. It appears that Ampere's law in its original form gives a contradiction. While it is true that no usual current can be measured between the capacitor plates, it is clear that an electric field is being built up as the capacitor is connected to a voltage source. The current associated with the displacement of charge can be calculated from the total charge deposited by the current in the wire. The electric field in the capacitor is homogeneous and the electric flux is related to the field strength, and the area:

Phi[e]=E*A=q/epsilon[0]

The displacement current is obtained from the time derivative:

I[d]=epsilon[0]* diff(Phi[e],t)

The corrected version of Ampere's law contains a sum of the usual current and the displacement current on the right-hand side.

We can examine the magnetic field between two circular capacitor plates as they are charged.

Suppose the plates have radius R , and we are interested in the magnetic field on a circle with radius R /2 (the magnetic field vectors are tangential to this circle).

We have for the magnetic flux

> B:='B':

> Phi[m]:=B*2*Pi*R/2;

The electric field is calculated from the time-varying charge q ( t ) and the plate area as:

> E:=q(t)/epsilon[0]/(Pi*R^2);

We need the electric flux through the area bounded by the circle with radius R /2:

> Phi[e]:=E*Pi*(R/2)^2;

> Id:=epsilon[0]*diff(Phi[e],t);

> Maxwell:=Phi[m]=mu[0]*Id;

> Bd:=solve(Maxwell,B);

We insert some numerical values:

> simplify(subs(diff(q(t),t)=1*A*s/s,mu[0]=4*Pi*10^(-7)*N/A^2,R=0.5*m,Bd));

This is the magnetic field strength in Tesla; note that the rate of charging was taken as 1 C/s=1 A.

We have learned in these lines the following:

1) for DC currents capacitors are considered to be insulators; this is not true during turn-on and turn-off, when the plates are being charged/discharged.

2) during the charge/discharge times of the plates there is current passing through the wires that connect to the plates, but not in-between the plates.

3) nevertheless, there is a magnetic field associated with the displacement of charge during the process of charging/discharging the plates.

4) for an AC current the capictor acts as a frequency-dependent resistor (impedance); still no current passes in-between the plates (we have a vacuum, or a dielectric between the plates, and we limit the voltages such that no discharges occur), but the displacement current causes a magnetic field according to Maxwell's generalization of Ampere's law.

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