30.3.1 Magnetic Flux in a Nonuniform Field

Equation 30.10 for the magnetic flux assumes that the field is uniform over the area of the loop. We can calculate the flux in a nonuniform field, one where the field strength changes from one edge of the loop to the other, but we’ll need to use calculus.

FIGURE 30.15 shows a loop in a nonuniform magnetic field. Imagine dividing the loop into many small pieces of area dA. The infinitesimal flux dΦm through one such area, where the magnetic field is B⃗, is

dΦm=B⃗⋅dA⃗ (30.11)

FIGURE 30.15 A loop in a nonuniform magnetic field.

The figure illustrates a circular loop in a non-uniform magnetic field.
FIGURE 30.15 Full Alternative Text

The total magnetic flux through the loop is the sum of the fluxes through each of the small areas. We find that sum by integrating. Thus the total magnetic flux through the loop is

Φm=∫B→⋅dA→area of loop (30.12)

Equation 30.12 is a more general definition of magnetic flux. It may look rather formidable, so we’ll illustrate its use with an example.

EXAMPLE 30.5 Magnetic flux from the current in a long, straight wire

The 1.0 cm×4.0 cm rectangular loop of FIGURE 30.16 is 1.0 cm away from a long, straight wire. The wire carries a current of 1.0 A. What is the magnetic flux through the loop?

FIGURE 30.16 A loop next to a current-carrying wire.

The figure illustrates a rectangular loop placed to the right of a long straight wire.

MODEL Model the wire as if it were infinitely long. The magnetic field strength of a wire decreases with distance from the wire, so the field is not uniform over the area of the loop.

VISUALIZE Using the right-hand rule, we see that the field, as it circles the wire, is perpendicular to the plane of the loop. FIGURE 30.17 redraws the loop with the field coming out of the figure and establishes a coordinate system.

FIGURE 30.17 Calculating the magnetic flux through the loop.

The figure illustrates rectangular loop on an x y coordinate axis.

SOLVE Let the loop have dimensions a and b, as shown, with the near edge at distance c from the wire. The magnetic field varies with distance x from the wire, but the field is constant along a line parallel to the wire. This suggests dividing the loop into many narrow rectangular strips of length b and width dx, each forming a small area dA=b dx. The magnetic field has the same strength at all points within this small area. One such strip is shown in the figure at position x.

The area vector dA⃗ is perpendicular to the strip (coming out of the figure), which makes it parallel to B⃗(θ=0°). Thus the infinitesimal flux through this little area is

dΦm=B⃗⋅dA⃗=B dA=B(b dx)=μ0Ib2πx dx

where, from Chapter 29, we’ve used B=μ0I/2πx as the magnetic field at distance x from a long, straight wire. Integrating “over the area of the loop” means to integrate from the near edge of the loop at x=c to the far edge at x=c+a. Thus

Φm=μ0Ib2π∫cc+adxx=μ0Ib2πln x|cc+a=μ0Ib2πln(c+ac)

Evaluating for a=c=0.010 m, b=0.040 m, and I=1.0 A gives

Φm=5.5×10−9 Wb

REVIEW The flux measures how much of the wire’s magnetic field passes through the loop, but we had to integrate, rather than simply using Equation 30.10, because the field is stronger at the near edge of the loop than at the far edge.