Hippocampus Physics: Ampere's Law

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Now consider a loop of wire of radius R, carrying a current I. What is the magnetic field at a point P located on the axis of the circular loop? We will use the Biot-Savart Law to solve this problem. In the loop, dx is perpendicular to the unit vector r. Therefore the cross product of dx and r is dx. The vector of the magnetic field dB is perpendicular to the plane and can be decomposed into two components, d B x and d B y. When we sum up all the elements of d B y around the loop, they cancel because of symmetry. This leaves only d B x equals d B cosine theta to be considered. Cosine theta is the radius divided by x squared plus r squared to the one-half power. By integrating the x component of the magnetic field, and using the fact that the integral of dx is the circumference of the loop, 2pi times the radius, we find an expression for the magnetic field strength at a point along the axis of a circular loop. To find the magnetic field at the center of the loop, we just set x equal to zero, and find that the magnetic field at this special point equals mu times the current divided by 2 times the radius. Far from the loop, which in this case means x greater than the radius, the magnetic field strength is equal to mu times the current times the radius squared divided by 2 times the distance cubed. Ampere came up with another way to look at the relationship between a current in a wire and the magnetic field associated with it. Consider an irregular loop that encloses a wire carrying a current coming out of the screen. This loop is called an Amperian Loop and may have any shape as long as it is a closed path and around the wire. The strength of the magnetic field acting on the loop is given by the closed integral of the dot product of B and d s. We use a closed integral for a closed path. The dot product represents the product of a segment, d s, of the Amperian loop and the magnetic field component that is tangent to the loop, B cosine theta. If the angle theta is equal to zero or 180 degrees, cosine theta is zero. Ampere's law states that the line integral of B dot ds around any closed path... equals mu zero times I, where I is the total current flowing through the closed path surface. Use Ampere's Law to find the magnetic field strength at a point P a distance, r, from a wire carrying a current, I. The closed integral of d s is simply the circumference of the loop, 2 pi times r. Substituting these values into the equation, we find that the total magnetic field at a... ...point P created by an infinitely long straight wire is mu zero times I divided by 2 pi times the distance r. Note how much simpler this approach is than using Biot-Savart's Law - and the results are the same! Let's use Ampere's law to find the magnetic field inside a long current-carrying wire of radius R. The current passing through the Amperian loop is less than the current passing through the wire. In addition, the current enclosed by the Amperian loop is proportional to the area enclosed by the Amperian loop. Therefore, the current enclosed by the loop divided by the area enclosed by the loop is equal to the current in the wire divided by the area of the wire. Using Ampere's law, we find the magnitude B of the magnetic field inside the wire: mu times the current divided by 2 pi times the radius of the wire squared times the radius of the amperian loop. What is the magnetic field outside a long current-carrying wire of radius R?