Ch05 Electromagnetic Induction and Alternating Current

PHYSICS 2ND PAPER CHAPTER 5 Electromagnetic Induction and Alternating Cur- rent Question 1 [Din. B. 2019] In a step-up transformer, an alternating electromotive force of 2200 V is obtained across the secondary coil when 220 V is supplied to the primary coil. The number of turns in the primary coil is 500 and its electrical resistance is 0.5 Ω. (a) What is a transformer? (b) Explain why a transformer cannot operate with direct current (DC). [D. B. 2015; S. B. 2016; Ctg. B. 2017] (c) Determine the number of turns in the secondary coil of the transformer. (d) According to the stem, can the ratio of the resistances of the primary and secondary coils be determined? Give your opinion with mathematical analysis. Solution (a) A transformer is a static device that changes an alternating voltage from one value to another (step up or step down) without changing its frequency. It works on the principle of mutual induction. (b) A transformer works by mutual induction. The emf induced in the secondary coil is 𝑒𝑠= −𝑁𝑠 dΦ d𝑡 where Φ is the flux in the core due to the primary current. With direct current (DC) in the primary: 1. The current does not change with time, so the flux in the core is const

For a transformer, the voltage ratio equals the turns ratio: 𝑁𝑠 𝑁𝑝 = 𝑉𝑠 𝑉𝑝 𝑁𝑠= 𝑁𝑝× 𝑉𝑠 𝑉𝑝 = 500 × 2200 220 = 5000 𝑁𝑠= 5000 (d) Yes, the ratio can be found if we take the usual design conditions of a transformer: 1. both coils are made of the same metal (same resistivity 𝜌); 2. each turn has about the same length, so the wire length is proportional to the number of turns, ℓ∝𝑁; 3. the wire of each coil is chosen so that the current per unit cross-section is the same, so the area of the wire is proportional to its current, 𝐴∝𝐼. For an ideal transformer, 𝑉𝑝𝐼𝑝= 𝑉𝑠𝐼𝑠. So 𝐼𝑝 𝐼𝑠 = 𝑉𝑠 𝑉𝑝 = 𝑁𝑠 𝑁𝑝 = 5000 500 = 10 Resistance of a wire is 𝑅= 𝜌ℓ 𝐴. So 𝑅𝑠 𝑅𝑝 = ℓ𝑠 ℓ𝑝 × 𝐴𝑝 𝐴𝑠 = 𝑁𝑠 𝑁𝑝 × 𝐼𝑝 𝐼𝑠 = 10 × 10 = 100 With 𝑅𝑝= 0.5 Ω: 𝑅𝑠= 100 × 0.5 = 50 Ω The secondary has 10 times more turns of a wire that is 10 times thinner in area. So its resistance is 100 times larger. Conclusion: yes, the ratio can be determined: 𝑅𝑝: 𝑅𝑠= 1 : 100. 𝑅𝑝: 𝑅𝑠= 1 : 100, 𝑅𝑠= 50 Ω ahbab.dev 2

Question 2 [Ctg. B. 2017] A planar coil of mean area 100 cm2 and having 200 turns is held perpendicularly to the direction of a uniform magnetic field of intensity 0.2 × 10−4 T. The coil is then rotated through 180◦ in a time interval of 0.1 s. (a) Give the definition of magnetic flux. [Din. B. 2019] (b) State Lenz’s law and show that it obeys the law of conservation of energy. [Ctg. B. 2015; All Boards 2018] (c) Determine the mean value of the electromotive force induced in the coil during this rotation. (d) Explain mathematically the nature and values of the induced current in the coil when it is rotated continuously through a complete cycle of 360◦with the same uniform speed. Solution (a) The magnetic flux through a surface is the number of magnetic field lines passing normally through that surface. Mathematically, Φ = ®𝐵· ®𝐴= 𝐵𝐴cos 𝜃 where 𝜃is the angle between ®𝐵and the normal to the surface. Its SI unit is the weber (Wb), where 1 Wb = 1 T m2. (b) Lenz’s law: the direction of the induced emf (and induced current) is always such that it opposes the change in magnetic flux that produces it. Lenz’s law and conservation of energy: 1. When the north pole of a magnet is pushed towar