Sources of the Magnetic Field

• Review of the Electric Field
• electric charges produced an electric field
• another electric charge would feel a force proportional to the field



• 



• The total electric field of a continuous distribution is the sum of the contributions from a number of infinitesimally small pieces



• 



• Magnetic Force
• a current carrying wire feels a force when it is in a magnetic field
• does a current carrying wire produce a magnetic field
• Oersted (1820)
• a compass needle placed by a wire will deflect when the current is turned on.
• two current carrying wires produce a force on each other
• Ampere developed an expression for the force



• 



• This can be grouped as



• 




• Biot-Savart Law allows us to determine the magnetic field for an arbitrary current distribution
• Examples
• Long straight wire
• parallel to the x axis, of length = x₁ + x₂,
• want to find the magnetic field at point P.









• q measures the angle that r makes with the positive x-axis, it will be less than 90^o for values of x < 0 and greater than 90^o for values of x greater than zero.
• produces a field in the direction pointing out of the paper (Use the right hand rule for the cross product)
• the magnitude of dB will be



• 



• our wire is in the x-direction and the magnitude of a cross product is equal to the product of the magnitude of the vectors multiplied by the sine of the angle between them.
• from the diagram, we can see that r, x, and q are all related. Before we can evaluate the integral to determine B, we need to express dB in terms of a common variable




• 




• substituting the above into our expression for dB




• 



• and the total magnetic field is



• 



• For an infinitely long wire, q₁ goes to zero and q₂ goes to i, and the magnetic field becomes
• 



• The magnetic field will form concentric loops around the wire.
• Curved Arc






• circular arc of radius R, want to find the magnetic field at the center of curvature.
• The straight segments of wire do not contribute to total magnetic field because the element of length is parallel to the displacement vector.
• along the curved segment, ds is perpendicular to r and sin q = 1. Using the right hand rule, the magnetic field will be directed into the page.



• The contribution from some small segment of length will be equal to



• ,



• to find the total magnetic field, we add up the contributions from each small segment,
• I and R are both constant over the curve, and the total magnetic field becomes



• , s is the length of the curve and is equal to R q



• , when the curve becomes a complete circle, q = 2 p and



• 



• Example - Circular Segment of Wire, on axis and out of plane

Magnetic Field of on the Axis of a Circular Current Loop

Let the z axis be parallel to the axis of the loop. The Biot-Savart law for one small section of the current carrying wire is

ds will always be perpendicular to r. (Demonstrated in class)

The direction of B is determined by using the right hand rule. The vector r makes an angle of q with the x-y plane. Because B is perpendicular to r, it will make an angle q with the z axis, which is perpendicular to the x-y plane. See diagram.

The magnitude of the contribution from the section of wire to the total magnetic field is

The vector can be resolved into a component parallel to the x-y plane and one perpendicular to the plane. If we look at the contributions from all of the sections, we can see that the components parallel to the x-y plane will sum to zero. The net magnetic field will be in the z direction. The component of dB that contributes to the total magnetic field will be the component that is parallel to the z axis.

The total magnetic field is the sum of the z components from all of the little pieces of the loop.

In the limit of z = 0, this reduces to the expression we had for the field at the center of a circular loop.