Household electricity in a particular country is supplied in the form of alternating current that varies from 180 V to −180 V with a frequency of 100 cycles per second (Hz). The voltage is thus given by the equation E(t) = 180 sin(200πt) where t is the time in seconds. Voltmeters read the RMS (root-mean-square) voltage, which is the square root of the average value of [E(t)]2 over one cycle. (a) Calculate the RMS voltage of household current in this particular country. (Round your answer to the nearest whole number of volts.) V (b) Many electric stoves require an RMS voltage of 220 V. Find the corresponding amplitude A needed for the voltage E(t) = A sin(200πt). (Round your answer to the nearest whole number of volts.) A = V

a. 127 V b 311 V

Explanation:

a. The RMS value of E(t) = 1/T∫[E(t)]². E(t) = 180 sin(200πt). Since the frequency f = 100 cycles per second, the period, T = 1/f = 1/100 = 0.01 s.

So, 1/T∫[E(t)]² = 1/T∫[180 sin(200πt)]² = 180²/0.01∫sin(200πt)]²

Using trigonometric identity sin²Ф = (1 - cos2Ф)/2 where Ф = ωt

1/T∫[E(t)]² = 180²/0.01∫(1 - cos2Ф)/2. We integrate from 0 to T, , we have

1/T∫[E(t)]² = 180²/(0.01 × 2)(t - sin2ωt/2ω)

1/T∫[E(t)]² = 180²/(0.02[(T - (sin2π)/(2 × 200π) ) - (0 - [sin(2 × 0)]/(2 × 200π))

1/T∫[E(t)]² = 180²/(0.02)[(0.01 - 0)

1/T∫[E(t)]² = 180²/2

E(t)RMS = √1/T∫[E(t)]²

= √180²/2

= 180/√2

= 127.28 V

= 127 V to the nearest whole number

b. Since E(t)(RMS) = A/√2 where A = voltage amplitude and E(t)(RMS) = 220 V,

A = √2E(t)(RMS) =

√2 × 220 V

= 311.13 V

= 311 V to the nearest whole number