Fields: Magnetic vs. Electric The two field types also have important differences: ･ An electric field is caused by any charge. A magnetic field is caused only by a moving charge. ･ Electric field lines originate at one point in space (a positive charge) and terminate at another point in space (a negative charge). Magnetic field lines form closed loops. ･ An electric field exerts a force on any charge within that field. A magnetic field exerts a force only on a charge that is moving within that field—with some component of its velocity perpendicular to the field lines. ･ An electric field exerts a force parallel to the field lines. A magnetic field exerts a force perpendicular to both the field lines and the charge’s velocity. 5/15/17 Oregon State University PH 213, Class #19

Magnetic Forces on Charges Compare: Magnitude Direction FE = q0E (parallel to E) Fmag = q0(v·sin)B (perpendicular to B and v) Why sin? What’s  in this equation? It’s the magnitude of the angle between v and B. 5/15/17 Oregon State University PH 213, Class #19

Oregon State University PH 213, Class #19 Fmag = q0(v·sin)B (perpendicular to B and v) But which perpendicular direction does Fmag have? Use Right-Hand Rule #1: Thumb in direction of v. Fingers in direction of B. Palm faces direction of Fmag on a positive q0. Note: Fmag is opposite on a negative q0; the back of the hand indicates its direction. We denote B-field lines in and out of the page as  and (). (Notice: The conventional orientation of the positive coordinate axes use RHR #1, too: The +x-axis is your thumb; the +y-axis is your fingers; the +z-axis comes up out of the page—your palm.) 5/15/17 Oregon State University PH 213, Class #19

Oregon State University PH 213, Class #19 An electron moves perpendicular to a magnetic field. The magnetic force that acts on the electron is shown. What is the direction of the magnetic field causing this force? Left Into the page Out of the page Up Down 5/15/17 Oregon State University PH 213, Class #19

Oregon State University PH 213, Class #19 Fmag = qvB·sin Find the magnetic force (both magnitude and direction) exerted on an electron traveling horizontally due south, at a speed of 105 m/s, through a uniform magnetic field (4 T) that is directed horizontally at 30° east of north. 1. 6.4 x 10-14 N downward 2. 6.4 x 10-14 N upward 3. 3.2 x 10-14 N downward 4. 3.2 x 10-14 N upward 5. None of the above. 5/15/17 Oregon State University PH 213, Class #19

The Motion and Energy of a Charge in a B-Field The magnetic force, Fmag, is always perpendicular to the velocity of the moving charge. Therefore, Fmag can act as a radial force—causing circular motion—when v and B are also perpendicular: Fmag = q(v·sin)B = qvB = FC = mv2/r where m is the mass of the particle with charge q. Conclusion: r = mv/(qB) 5/15/17 Oregon State University PH 213, Class #19

Oregon State University PH 213, Class #19 Example: A proton is traveling horizontally east at a steady speed of 3 x 106 m/s when it enters a region with a uniform magnetic field directed downward (i.e. into the earth). Describe the resulting motion of the proton. What if the particle were an electron rather than a proton? 5/15/17 Oregon State University PH 213, Class #19

The Source of the Magnetic Field: Moving Charges The magnetic field of a charged particle q moving with velocity v is given by the Biot-Savart law: where r is the distance from the charge, θ is the angle between v and r, and µ0 is a universal constant (= 4p x 10–7 T·m/A). The Biot-Savart law can be written as cross product: 5/15/17 Oregon State University PH 213, Class #19

Oregon State University PH 213, Class #19 Example: A proton is traveling in the +x-direction at a steady speed of 3 x 106 m/s. Describe the magnetic field it is causing at the points (0, 5), (4, 3) and (4, –3) as it passes through the origin. All coordinates here are given in meters. What if this were an electron rather than a proton? 5/15/17 Oregon State University PH 213, Class #19