Calculating the Magnetic Field Due0 to a Current
As we discussed in Section 29-1, one way to produce a magnetic field is with moving charges-that is, with a current. Our goal in this chapter is to calculate the magnetic field that is produced by a given distribution of currents. We shall use the same basic procedure we used in Chapter 23 to calculate the electric field produced by a given distribution of charged particles. Let us quickly review that basic procedure. We first mentally divide the charge distribution into charge elements do, as is done for a charge distribution of arbitrary shape in . We then calculate the field if produced at some point P by a typical charge element. Because the electric fields coil.used by different elements can be superimposed, we calculate the net field f at P by summing. via integration, the contributions if from all the elements. Recall that we express the magnitude of do as.
which indicates that the direction of the vector dE produced by a positively charged element is the direction of the vector r. Note that Eq. 30-2 is an inverse-square law(dE depends on inverse r2) in spite of the exponent 3 in the denominator. That exponent is in the equation only because we added a factor of magnitude r in the numerator. Now let us use the same basic procedure to calculate the magnetic field due to a current. shows a wire of arbitrary shape carrying a current i. We want to find the magnetic field B at a nearby point P. We first mentally divide the wire into differential elements ds and then define for each element a length vector d’S that has length ds and whose direction is the direction of the current in ds. We can then define; differential current-length element to be I dS; we wish to calculate the field’ dB produced at P by a typical curry not-length element. From experiment we find that magnetic fields, like electric fields, can be superimposed to find a net field. Thus, we can calculate the net field B at P by summing, via integration, the contributions dB from all the current-length elements. However, this summation is more challenging than the process associated with electric fields because of a complexity; whereas a charge element dq producing an electric field is a scalar, a current-length element i d’S producing a magnetic field is the product of a scalar and a vector. The magnitude of the field dB produced at point P by a current-length element dS turns out to be
Magnetic Field Due to a Current in a Long Straight Wire
Shortly we shall use the law of Biot and Savart to prove that the magnitude of the magnetic field at a perpendicular distance R from a long (infinite) straight wire carrying a current i is given by.