1. 1 15. Magnetic field Historical observations indicated that certain materials attract small pieces of iron. In 1820 H. Oersted discovered that a compass needle will orient itself perpendicularly to a current - carrying wire. In a neighborhood of a permanent magnet or a current- carrying wire there exists a field, called the magnetic field B. If we wind a wire in a form of a helical coil (a solenoid), one obtains much higher magnetic field. Magnetic field lines for a bar magnet Magnetic field of a straight current- carrying wire. The direction of a magnetic field can be found from the corkscrew rule Magnetic field lines for a streched-out solenoid. The field is strong and uniform at interior points. From HRW 3

2. 2 15.1. Force in a magnetic field A magnetic field B is defined in terms of the magnetic force exerted on an electric test charge q 0 moving with velocity v, using the vector equation (15.1) The SI unit for B is the tesla (T) 1T = 1N/(A· m) = 1 Wb/m 2 In a space where both fields B and E exist, the force acting on a charged particle is (15.2) The electric current is formed by mobile charge carriers, hence in a magnetic field a force should act also on a current carrying wire. The force acting on dq, according to (15.1), is (15.3) From the definition of a current we have (15.4) Substituting (15.4) into (15.3) one gets for a force acting on a segment of a current- carrying wire (15.5) In an element dl of a wire there exists a charge dq

3. 3 Force in a magnetic field, cont. Integrating eq.(15.5) one obtains for a force acting on a current-carrying wire of length L placed in the field B (15.6) Sample problem What is the force acting on a segment L of a wire carrying current I, which makes an angle θ with a uniform magnetic field B. From (15.6) one gets For θ = π/2 F = B I L For θ = 0 F = 0 The direction of dl is that of the current I

4. 4 15.2. Amper’s law Similarly to the calculation of electric field from a Gauss’ law, in cases where the distribution of magnetic field has some symmetry, one can apply Amper’s law to find B (15.7) Magnetic field outside a long straight wire To calculate the field we use Amper’s law. Vector B is tangent to the loop and has a constant magnitude along the loop. (15.8) The line integral of the magnetic field B along any closed path is equal to the total current enclosed inside the path multiplied by permeability constant μ 0 Amperian loop L reflects the cylindrical symmetry of the field

5. 5 Amper’s law, cont. Example What is the force of attraction between two parallel wires carrying currents I 1 and I 2 in the same direction. We assume that a wire with current I 2 is placed in the magnetic field produced by a wire with current I 1. The force acting on a wire of length L in the field B is The magnitude of B is The force acting from a wire with current I 1 on a unit of length of a wire with current I 2 is then (15.9) Eq.(15.9) is used for defining a unit of current in SI units, the ampere (A). Putting in this equation I 1 = I 2 = 1A and d = 1m one obtains: The ampere is that constant current which, if maintained in two straight, parallel conductors of infinite length, of negligible circular cross section, and placed 1m apart in vacuum, would produce on each of these conductors a force of magnitude 2 x 10 -7 newton per meter of wire length.

6. 6 Amper’s law, cont. Magnetic field of a solenoid We take into consideration an ideal solenoid, which is infinitely long and consists of close-packed turns. The field inside the coil is uniform and parallel to the axis, the field ouside is zero. The sense of B can be determind using the right-hand rule: if the fingers indicate the direction of current in a coil, the thumb of the right hand points along B. To determine the magnetic field inside a solenoid we write Amper’s law as a sum of integrals for each loop segment The first integral ( field B is uniform ) For the rest of integrals ( B either is perpendicular to dl or is zero ) Amper’s law then gives(15.10) where N is a number of turns enclosed by the loop and n a number of turns per unit length. The obtained result agrees with experimental observations that the magnetic field inside a solenoid does not depend on the diameter or the length of the solenoid. Outside view of a real solenoid A section of an ideal solenoid with the rectangle loop abcda used in integration From HRW 3

7. 7 Amper’s law, cont. An example of an electromagnet used in experiments with the deflection of particles is shown in the figure below. Some parameters of the electromagnet: field in the slit (60 x 20 cm) B = 16 000 Gs = 1.6 T height 180 cm width 220 cm keeper of a magnet: Armco iron number of turns 2N = 1310 cross section of the wire 1,15 cm 2 power used 82,5 kW. Outside view (a) and a cross section (c) of a real electromagnet

8. 8 15.3. Biot - Savart’s law The Biot-Savart’s law is used for determination of a magnetic field in situations of the arbitrary distribution of current. In particular this law gives the field dB generated by a wire segment dl that carries a current I. (15.11) Biot – Savart’s law The magnitude of vector dB is (15.12) The field generated by a wire of length L is found by integration of eq.(15.12) (15.13) Sample problem Determine the magnetic field produced by a segment of a straight wire carrying a current I. From the Biot-Savart’s law one gets: (15.14) A current-length element I dl produces the field dB at point P given by a vector r connecting dl and P.

9. 9 Sample problem, cont. From the figure it follows that Introducing new variables one obtains Eq.(15.14) can then be written as (15.15) From (15.15) it follows that for a wire of infinite length, i.e. for θ 1 = 0, θ 2 = π, we get what agrees with the result (15.8) obtained from the Amper’s law. The field is calculated at a point seen from both ends of a segment at angles θ 1 and θ 2

10. 10 16. Faraday’s law of induction In a series of experiments M. Faraday have shown that it is possible to generate electric currents without the use of batteries. Some of these experiments are shown below In above experiments the cause of induced current and an emf is the change in a magnetic flux through the loop. Faraday’s law (15.16) A current is flowing in a coil when the magnet is moving loop 1 loop 2 An ammeter registers a current in a loop when the switch S is opened or closed. The loops are at rest. The magnitude of induced emf in a loop is equal to the rate of change in a magnetic flux through the loop. ΦBΦB From HRW 3

11. 11 Faraday’s law of induction, cont. If in the induction experiments we use instead of a loop the coil with N turns, the emf will also be N times larger: (15.17) The sign „-” in eqs(15.16) and (15.17) reflects the Lenz’s rule: This rule can be inferred from the law of energy conservation. The external agent must always do work on the loop-magnetic field system. This work appears as the thermal energy dissipated on the resistance of the loop wire. A the magnet approaches the loop, the induced current produces the magnetic field, which opposes the motion of the magnet. In opposite case the magnet would be attracted by the loop and a kind of perpetuum mobile would be formed. An induced current produces a magnetic flux which opposes the change of flux inducing the current. From HRW 3

12. 12 Examples of induction 1.A rectangular loop is pulled out of a magnetic field With the external force we pull out a loop with a velocity v. The magnetic flux through the loop is Emf induced in a loop can be calculated from the Faraday’s law The current flowing in a loop is 2. A loop rotating in a magnetic field A loop of area S is rotating in a constant field B with angular frequency ω. The magnetic flux and emf are For N turns (a coil) the magnitude of ε 0 is N times larger. This is an example of the commercial alternating current generator. In this case ω=2π · 50 Hz.

13. 13 Examples of induction, cont. 3. A changing magnetic field The changing magnetic field gives the changing magnetic flux. The induction current flows under the influence of an electric field. This electric field is induced even if there is no copper ring enabling flowing of a current. The electric field lines produced by the changing magnetic field form a set of concentric circles. With the induced field E an emf can be combined. The work done by emf during moving a test charge around the circle is what gives In a general case of work done along any closed path The induction law can then be written as This general form of Faraday’s law is one of the Maxwell’s equations.