Physics 272 March 11 Spring 2014 http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html Prof. Philip von Doetinchem philipvd@hawaii.edu Phys272 - Spring 14 - von Doetinchem - 32
Summary Magnetic field of a current element (vector version): Law of Biot and Savart: Important law to find the total magnetic field of at any point in space due to currents Phys272 - Spring 14 - von Doetinchem - 33
Summary Two protons moving in the same direction. Magnetic force on the upper proton from the magnetic field of the lower proton: attractive Magnetic force on the lower proton from the magnetic field of the upper proton: attractive Phys272 - Spring 14 - von Doetinchem - 37
Force between moving charges http://www.youtube.com/watch?v=43aeudvwc0k Moving in the same direction for likewise signed charges creates an attractive force and moving in the opposite direction creates a repulsive force Phys272 - Spring 14 - von Doetinchem - 38
Force between parallel conductors Interaction force between conductors Important when wires are close to each other Force on wire 1 due to magnetic field from wire 2: Phys272 - Spring 14 - von Doetinchem - 39
Magnetic forces and defining the Ampere Definition of the SI unit for current Ampere One ampere is that unvarying current that, if present in each of two parallel conductors of infinite length and one meter apart in empty space, causes each conductor to experience a force of exactly 2x10-7N per meter of length. Definition of 1 Coulomb based on the same approach: 1C=1A/s Phys272 - Spring 14 - von Doetinchem - 41
Magnetic field of a circular current loop Coils of wire with a large number of turns spaced closely Each turn is nearly planar Current in such a coil causes a magnetic field http://www.youtube.com/watch?v=7wmzru6vqze Look at one loop first then study the case of multiple turns Phys272 - Spring 14 - von Doetinchem - 42
Magnetic field on the axis of a coil Magnetic field on axis of the loop along the x axis at a certain distance x using Biot-Savart Phys272 - Spring 14 - von Doetinchem - 43
Magnetic field on the axis of a coil All magnetic field contributions in the plane of the loop cancel out and only a component along the loop axis remains Phys272 - Spring 14 - von Doetinchem - 44
Magnetic field on the axis of a coil Integration along the circular loop B field along the axis of the loop: Phys272 - Spring 14 - von Doetinchem - 45
Magnetic field on the axis of a coil Stack a number of N loops on top of each other Assume that each loop is planar and that the loops are very closely spaced Also assume that the distance from the center of each loop is the same to the point along the coil axis under study Superposition principle at work: Many loops are able to produce very strong fields easier than having high currents Phys272 - Spring 14 - von Doetinchem - 46
Magnetic field on the axis of a coil Express with the help of magnetic dipole moment: Magnetic dipole moment does not only react to magnetic field, but also acts a source of magnetic field in this sense Be careful: equations on last 4 slides are only valid along the symmetry axis of the coil Phys272 - Spring 14 - von Doetinchem - 47
Magnetic field of a coil A coil with 100 circular loops with radius 60cm carries a current of 5.0A: Phys272 - Spring 14 - von Doetinchem - 48
Magnetic field of a coil A coil with 100 circular loops with radius 60cm carries a current of 5.0A: Phys272 - Spring 14 - von Doetinchem - 49
Ampere's law Magnetic field calculation so far based on adding contribution of small line segments analogous to electric field calculation with adding up electric fields of individual charges Gauss's law was a different way to look at the problem from the outside in symmetrical cases Gauss's law for magnetism does not allow to make conclusions on current magnetic flux through enclosed surface is always zero Phys272 - Spring 14 - von Doetinchem - 50
Ampere's law Integrate along a closed path around a wire: Phys272 - Spring 14 - von Doetinchem - 51
Ampere's law Integrate along a closed path around a wire: Phys272 - Spring 14 - von Doetinchem - 52
Ampere's law Integrate along a closed path, but not totally enclosing the wire: If the current is not enclosed the line integral around a closed path is zero Phys272 - Spring 14 - von Doetinchem - 53
Ampere's law: general statement The scalar product always takes the parallel projection for the integration as long as the closed path integration encloses the current we get: This also true if multiple conductors are present within the closed integration path (magnetic field is the sum of the individual magnetic field of the different conductors) This law is true for steady currents time variation next lectures Important: does not mean that the magnetic field is zero Phys272 - Spring 14 - von Doetinchem - 54
Important differences electric field vs. magnetic field Electric field: Line integral along a closed path is zero Electric force is conservative force does zero net work on charge that returns to start point Magnetic field: Magnetic force on a moving charge is always perpendicular line integral over magnetic field is not related to the work done by magnetic force Magnetic force is not conservative force does not only depend on position, but also on velocity Phys272 - Spring 14 - von Doetinchem - 55
Application of Ampere's law Ampere's law useful for problems with symmetries to determine the magnetic field To calculate magnetic field at a certain point the integration path has to go through this point Look for symmetries to make the analytical solution of the integral possible Integration path should be tangent in regions of interest and perpendicular in other regions (makes the scalar product easy) No net current enclosed: line integral is zero Make sanity checks: how does the calculated field behaves in different limits Phys272 - Spring 14 - von Doetinchem - 56
Field of a long, straight, current-carrying conductor Similar to the example before, but reversed situation: magnetic field unknown, current known Identify symmetry: magnetic field is tangent to a circle around the conductor Magnetic field magnitude is the same everywhere on the circle Phys272 - Spring 14 - von Doetinchem - 57
Field of a long cylindrical conductor Cylindrical conductor with radius R carries a current I. Current is uniformly distributed over the crosssectional area of the conductor. conductor Phys272 - Spring 14 - von Doetinchem - 58
Field of a long cylindrical conductor Phys272 - Spring 14 - von Doetinchem - 59
Field of a solenoid Helical winding of wire on a cylinder Each winding can be treated as circular loop All turns carry the same current Field is very uniform in the middle External field near the middle is very small Assumption: magnetic field zero outside and uniform inside Field is the strongest in the center Phys272 - Spring 14 - von Doetinchem - 60
Field of a solenoid Phys272 - Spring 14 - von Doetinchem - 61
Field of a solenoid Phys272 - Spring 14 - von Doetinchem - 62