Question

Question 4 A scheme for power generation proposes to use conducting balloons that expand and contract in waterfalls (with adequate insulation). Consider a balloon of volume V at a given time, with water flowing into it with a flow rate F(m/s). The water falls down into the spherical balloon causing it to expand. The balloon is in a region where the magnetic field is uniform and vertical of magnitude B. a) Calculate the proposed induced emf along the equator of the balloon. (horizontal equator), at the moment indicated above. (8 marks) b) Indicate the direction of flow of the current around the equator if the field B points down. (2 marks) c) The scheme then allows for the mouth of the balloon to be closed and water to leak out through holes in the bottom when the volume of the balloon becomes 4V. The initial leak rate is 2.5 F. calculate the induced emf along a horizontal ring on the balloon a vertical distance z away from the centre of the balloon. (6 marks) d) Calculate the induced emf along the largest vertical circle on the balloon for c). (4 marks)

Answer 1

1. a) flow rate = Rate of change of - volume of balloon. or FESY Volume V = 45183 om, aby zabky 1739) = 4 Tabiple en or, f = 498² do from Lena's law, induced emf due to change in flux is E = -d 5-(8. A) 5 º magnetic flux BiA is dot product or, E 2-P(BALOS0°)-(BA) 5 BX Ñ in same L direction, s As B is uniform (only à changes. - - =-Bd (3782) =- Bit.ar.do it A=T820 for feed Putting for at from E = =BH.RX. F = BF - © At this moment volume=v Ovi 491? = V 28. 244 or, y3 = 34 yos 13 Putting in 6 q = - B eu le

b) If seen from above, magnetic field pointing downward will be inside the plane of equator From Flemings right handoule. - Current .: Motion (Exponding) So direction of flow of current will be anti-clockwise when viewed from above. :) Flow rate = Change in volume of balloon Ox, -25% = () os, -2.5F = 4,50. Brzda Or, do = - 2.5f- from figure, radius of circle at distance z from Emp 2 --dm-- (BA) centre centre or, Ę = - B.dAz --B 4 (37(0-2)) Az = 5782 = 5T(82-22) --8..20 do Putting do from 6, *** & 5-8.5** - 2018 - 1 458 82=572_zz -2. 5F = 2.5BF 2. 28 239X STERS

At this moment volume = 4V 1 or 4 ST 83 = 41 or, 73 = 47x3 455 or, r=13v113 Patting in 6. ( T ) 13 2.5 BF : 2.5 BF XT 2x13 V V3 2 3 from Centre. This is the induced emf.at distance d) Largest vestical circle will be as shown Emf =-=-. (8) on LA 3-D (B. A cos 90°) dt As Area vector and magnedic field make angle goo with each other and cos 90°=0 So, £ 6 4 - 5