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Electromagnetism Faraday & Maxwell Faraday Michael Faraday (1791 – 1867) was an English scientist. He was a genius at experimental design and conceptualization (and an incompetent mathematician). Observations • • • • • Holding a magnet in front of a coil of wires does not induce a current. Pushing the north pole of a magnet through the coil induces a current in one direction. Pulling the north pole of a magnet back out of the coil induces a current in the other direction. The bigger the area of the coil, the larger the current. Twisting a magnet in front of a coil of wires causes a current only while the magnet is moving. Rutgers: example Induction A changing magnetic field around a coil of wires induces a potential difference, . Investigate this simulation. Induction A changing magnetic environment around a coil of wires induces a potential difference, . Strength of magnet voltage Area of coil voltage Number of coils voltage Rate of change of magnetic field voltage 𝜀𝛼𝐵 𝜀𝛼𝐴 𝜀𝛼𝑁 1 𝜀𝛼 ∆𝑡 Flux In physics, when we talk about “flux”, we refer to the number of things passing through or hitting a certain area. • • • Number of tennis balls landing in a particular area on the tennis court Number of photons hitting your retina Number of magnetic field lines passing through a particular area. Flux For purposes of electromagnetism, magnetic flux refers to the strength of magnetic field times the area: 1 tesla 1 square meter = 1 weber 1 W = 1 T 1 m2 We use the Greek letter phi (pronounced ‘fee’ or ‘fie’) subscript B to denote magnetic flux, B Wilhelm Weber, German, 1804 – 1891: investigated electricity and magnetism, co-invented telegraph. More here Changing flux: 1) With magnets of different strength B results in denser field lines and B Changing flux: 2) With different sizes of coils A results in more field lines and B Changing flux: 3) At different angles results in more field lines and B At different angles (cont’d) B B where B = component of magnetic field perpendicular to coil = B (cos ) B B (cos ) Changing flux: summary 𝐵 = 𝐵 𝐴 cos Where B = strength of magnetic field A = area of coil cos = angle between normal (face of coil) and direction of magnetic field How do we change induced current? A current is induced when there is a change in the magnetic flux through the loop’s area. B t N This relationship is called Faraday’s Law (but was articulated by James Maxwell) =𝑁 ∆𝐵 ∆𝑡 = ∆(𝐵 𝑐𝑜𝑠𝜃 𝐴) 𝑁 ∆𝑡 Example A square loop of wire with side length, l = 5.0 cm is in a uniform magnetic field, B = 0.16 T. What is the magnitude of the magnetic flux in the loop when B is perpendicular to the face of the loop? −2 𝑚 2 (cos 0°) Try it first 𝐵 =Try𝐵it first 𝐴 cos = 0.16 𝑇 5𝑥10 𝐵 = 4𝑥10−4 𝑇Try𝑚it 2first= 4𝑥10−4 W Example A square loop of wire with side length, l = 5.0 cm is in a uniform magnetic field, B = 0.16 T. What is the magnitude of the magnetic flux in the loop when B is at an angle of 30 to the face of the loop? Try it −2 first𝑚 2 (cos 30°) 𝐵 =Try𝐵it first 𝐴 cos = 0.16 𝑇 5𝑥10 𝐵 = 3.5𝑥10−4 W Try it first Example Suppose you rotate a square coil of wires with side length 5.0 cm in a magnetic field of strength 0.16 T. If it takes the coil 0.14 s to go from being perpendicular to the field to 30, how much current flows? Assume a resistor of 0.012 . 𝑉 𝑉 = Try 𝐼𝑅it𝑠𝑜 first𝐼 = 𝑅 ∆(𝐵 𝑐𝑜𝑠𝜃 𝐴) Where 𝑉 = 𝜀 Try = it first ∆𝑡 So, 𝐼 = 𝐵∆ 𝑐𝑜𝑠𝜃 𝐴 Try it first 𝑅 ∆𝑡 −2 𝑚 0.16 𝑇 𝑐𝑜𝑠0° − cos 30 ° 5 𝑥 10 Try it first 𝐼= (0.012 ) (0.14 𝑠) 𝐼 = 0.030 𝐴 Try it first 2 “Motional” Suppose you had a conducting bar connecting two parallel wires length L apart that ran perpendicular to an external magnetic field. If you were to slide the bar some distance x, you would increase the area and induce a current. 𝐵 = 0, so cos = 1 𝑁 = 1 𝐴 = 𝐿𝑥 𝑚𝑜𝑡𝑖𝑜𝑛𝑎𝑙 𝐵(𝑐𝑜𝑠𝜃)𝐴 ∆𝑥 =𝑁 = 𝐵𝐿 = 𝐵𝐿𝑣 ∆𝑡 ∆𝑡 Example Since blood contains ions, we can infer speed of blood flow if we place a test subject in a known magnetic field and measure the resulting current. Suppose a blood vessel is 2.0 mm in diameter, the magnetic field is 0.080 T, and the measured emf is 0.10 mV. What is the flow velocity of blood? Example Suppose a blood vessel is 2.0 mm in diameter, the magnetic field is 0.080 T, and the measured emf is 0.10 mV. What is the flow velocity of blood? it first so 𝑣 = =Try𝐵𝐿𝑣 𝑣= 𝑚 Try it first 0.63 𝑠 𝐵𝐿 = 0.10 𝑥10−3 𝑉 Try it first 0.080 𝑇 2.0 𝑥10−3 𝑚 Applications Generators Animal navigation Solid waste recycling Burglar alarms Metal detectors Speakers Seismometers Ground-fault circuit interrupters Computer memory Inductive chargers Etc. Transformers • It is more efficient to carry electricity long distances at high voltage. • However, high voltage can be very dangerous so is less useful in most situations. • A transformer is a device for increasing or decreasing the voltage of an alternating current (ac). Understanding transformers Alternating current in primary coil results in changing magnetic field in iron core Changing magnetic field propagates through iron core. Changing magnetic field induces current in secondary coil Physics of transformers 𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦 = 𝑁𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦 ∆𝐵 where = ∆𝑡 So, 𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦 𝜀𝑝𝑟𝑖𝑚𝑎𝑟𝑦 = ∆𝐵 ∆𝑡 𝜀𝑝𝑟𝑖𝑚𝑎𝑟𝑦 𝑁𝑝𝑟𝑖𝑚𝑎𝑟𝑦 𝑁𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦 𝑁𝑝𝑟𝑖𝑚𝑎𝑟𝑦 More… Efficient transformers are typically more than 99% efficient, so 𝑃𝑝𝑟𝑖𝑚𝑎𝑟𝑦 = 𝑃𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦 So, (𝐼𝑉)𝑝𝑟𝑖𝑚𝑎𝑟𝑦 = (𝐼𝑉)𝑠𝑒𝑐𝑜𝑛𝑑𝑎𝑟𝑦 Or 𝐼𝑠 𝑉𝑝 = 𝐼𝑝 𝑉𝑠 Example The charger for a cell phone contains a transformer that reduces 120-V alternating current to 5.0 V ac. Suppose the secondary coil contains 30 turns and the charger supplies 700 mA. 𝑉𝑠 𝑉𝑝 = 𝑁 , so 𝑁𝑝 Try𝑠it first 𝑁𝑠 30 𝑁𝑝 = = Try it first Try it𝑉 first 5 𝑠 120 𝑉𝑝 Try it first = 720 𝑡𝑢𝑟𝑛𝑠 How many turns in the primary coil? Example The charger for a cell phone contains a transformer that reduces 120-V alternating current to 5.0 V ac. Suppose the secondary coil contains 30 turns and the charger supplies 700 mA. What is the current in the primary coil? 𝐼𝑃 𝐼𝑠 𝑁𝑠 = Try it ,first so 𝑁𝑝 𝑁𝑠 30 Try it first 𝐼 = Try it first 𝐴) 𝐼𝑝 = (0.70 𝑠 𝑁𝑝 720 = 29 𝑚𝐴Try it first Example The charger for a cell phone contains a transformer that reduces 120-V alternating current to 5.0 V ac. Suppose the secondary coil contains 30 turns and the charger supplies 700 mA. How much power is transformed? it first 𝑃 =Try𝐼𝑉 Try it first 𝑃 = 0.70 𝐴 5𝑉 = 3.5 𝑊Try it first Typical power lines Lenz’s Law Changing magnetic fields lead to induced currents: ∆𝜑𝐵 → 𝜀 Those induced currents lead to induced magnetic fields. 𝜀→𝐵 Those induced magnetic fields resist further change to magnetic flux. 𝐵 →↓ ∆𝜑𝐵 If you want to induce currents, you have to do work. ∆𝐵 = −𝑁 ∆𝑡 Electromagnetic waves A changing electric field can induce a magnetic field. A changing magnetic field can induce an electric field. Therefore, it should be possible to create a self-sustaining electric and magnetic field independent of charges or currents! A changing electric field creates a magnetic field which then changes in just the right way to recreate the electric field which changes to recreate the magnetic field… etc. These are electromagnetic waves (more about which to come).