In 1819, Oersted discovered that a magnetic field is created around a wire carrying an electric current. In response to this discovery, two scientists - Jean Baptiste Biot and Felix Savart - began investigating this phenomenon. Their work yielded an expression that gives the magnetic field at a point in space in terms of the current. This expression is now called the Biot-Savart Law. Here is an example of the Biot-Savart Law which demonstrates the physical interpretation of the law. Imagine a horizontal wire that is carrying current to the right. Call the point at the left end of the wire, O, and choose another point, P, above the middle of the wire. Biot-Savart law gives the magnetic field produced by a small element of current-carrying wire at a point P. Use the right-hand rule to confirm that the magnetic field above the wire and at point P is directed out of the page and that its direction is into the screen below the wire. The magnetic field, B, at point P is perpendicular to the current, I. The direction of the magnetic field and the current are both perpendicular to r. It is important to understand that the Biot-Savart law gives the magnetic field at a point only for a small element in the wire. The strength of B at point P is the sum of all the contributions of the current along the entire length of the wire. The Biot-Savart law says that the strength of the magnetic field at P is... Directly proportional to the strength of the current, I, and the short segment of wire, dx. -Inversely proportional to the square of the distance, r squared. -Proportional to the sine of the angle between dx and r, and hence the cross product of dx and r. Construct the triangle as shown. Let's use the Biot-Savart law to find the magnetic field strength at a point P above a wire. The idea is first to express x and r in terms of the angle, theta. Using some simple geometry and algebra, we obtain the simple relationship between these variables: r equals y times the cosecant of theta. Using this relation we can express dx, as dx equals y times the cosecant of theta squared times d theta. Let's substitute this new expression for dx into the Biot-Savart Law. The expression for dB reduces to mu times I divided by 4 times pi times y; times sine theta d theta. Here the Biot-Savart Law is written to show the magnetic field strength, dB, at a point P of a really short segment of wire, dx, which is carrying a steady current, I. The end result is simply that the bit of the magnetic field at point P that is caused by the current in the small element of the wire below point P, is directly... ...proportional to the angle times a constant. The constant equals mu zero times the current divided by the quantity 4 pi y. Recall that Biot-Savart law gives the magnetic field produced by a small element of current-carrying wire at a point P. Therefore, to find the total magnetic field created by a wire of infinite length at some point P, we must integrate the Biot-Savart law expression we just found, over the entire range of elements from 0 to pi. By integrating both sides, we find that the total magnetic field at a point P created by an infinitely long straight wire is given by mu times... ...I divided by 2 pi times the perpendicular distance from point P to the wire, y. What is the direction and strength of a magnetic field 15 centimeters above a wire that is carrying 10.0 amperes of current to the left?

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