Given Information:
Resistance = R = 14 Ω
Inductance = L = 2.3 H
voltage = V = 100 V
time = t = 0.13 s
Required Information:
(a) energy is being stored in the magnetic field
(b) thermal energy is appearing in the resistance
(c) energy is being delivered by the battery?
Answer:
(a) energy is being stored in the magnetic field ≈ 219 watts
(b) thermal energy is appearing in the resistance ≈ 267 watts
(c) energy is being delivered by the battery ≈ 481 watts
Explanation:
The energy stored in the inductor is given by
![U = \frac{1}{2} Li^{2}](https://tex.z-dn.net/?f=U%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20Li%5E%7B2%7D)
The rate at which the energy is being stored in the inductor is given by
![\frac{dU}{dt} = Li\frac{di}{dt} \: \: \: \: eq. 1](https://tex.z-dn.net/?f=%5Cfrac%7BdU%7D%7Bdt%7D%20%3D%20Li%5Cfrac%7Bdi%7D%7Bdt%7D%20%5C%3A%20%5C%3A%20%5C%3A%20%5C%3A%20eq.%201)
The current through the RL circuit is given by
![i = \frac{V}{R} (1-e^{-\frac{t}{ \tau} })](https://tex.z-dn.net/?f=i%20%3D%20%5Cfrac%7BV%7D%7BR%7D%20%281-e%5E%7B-%5Cfrac%7Bt%7D%7B%20%5Ctau%7D%20%7D%29)
Where τ is the the time constant and is given by
![\tau = \frac{L}{R}\\ \tau = \frac{2.3}{14}\\ \tau = 0.16](https://tex.z-dn.net/?f=%5Ctau%20%3D%20%5Cfrac%7BL%7D%7BR%7D%5C%5C%20%5Ctau%20%3D%20%5Cfrac%7B2.3%7D%7B14%7D%5C%5C%20%5Ctau%20%3D%200.16)
![i = \frac{110}{14} (1-e^{-\frac{t}{ 0.16} })\\i = 7.86(1-e^{-6.25t})\\\frac{di}{dt} = 49.125e^{-6.25t}](https://tex.z-dn.net/?f=i%20%3D%20%5Cfrac%7B110%7D%7B14%7D%20%281-e%5E%7B-%5Cfrac%7Bt%7D%7B%200.16%7D%20%7D%29%5C%5Ci%20%3D%207.86%281-e%5E%7B-6.25t%7D%29%5C%5C%5Cfrac%7Bdi%7D%7Bdt%7D%20%3D%2049.125e%5E%7B-6.25t%7D)
Therefore, eq. 1 becomes
![\frac{dU}{dt} = (2.3)(7.86(1-e^{-6.25t}))(49.125e^{-6.25t})](https://tex.z-dn.net/?f=%5Cfrac%7BdU%7D%7Bdt%7D%20%3D%20%282.3%29%287.86%281-e%5E%7B-6.25t%7D%29%29%2849.125e%5E%7B-6.25t%7D%29)
At t = 0.13 seconds
![\frac{dU}{dt} = (2.3) (4.37) (21.8)\\\frac{dU}{dt} = 219.11 \: watts](https://tex.z-dn.net/?f=%5Cfrac%7BdU%7D%7Bdt%7D%20%3D%20%282.3%29%20%284.37%29%20%2821.8%29%5C%5C%5Cfrac%7BdU%7D%7Bdt%7D%20%3D%20219.11%20%5C%3A%20watts)
(b) thermal energy is appearing in the resistance
The thermal energy is given by
![P = i^{2}R\\P = (7.86(1-e^{-6.25t}))^{2} \cdot 14\\P = (4.37)^{2}\cdot 14\\P = 267.35 \: watts](https://tex.z-dn.net/?f=P%20%3D%20i%5E%7B2%7DR%5C%5CP%20%3D%20%287.86%281-e%5E%7B-6.25t%7D%29%29%5E%7B2%7D%20%5Ccdot%2014%5C%5CP%20%3D%20%284.37%29%5E%7B2%7D%5Ccdot%2014%5C%5CP%20%3D%20267.35%20%5C%3A%20watts)
(c) energy is being delivered by the battery?
The energy delivered by battery is
![P = Vi\\P = 110\cdot 4.37\\P = 481 \: watts](https://tex.z-dn.net/?f=P%20%3D%20Vi%5C%5CP%20%3D%20110%5Ccdot%204.37%5C%5CP%20%3D%20481%20%5C%3A%20watts)
The sound wave would behave differently in a swimming pool than in his bedroom because sound waves travel faster in more dense mediums; such as water. The wave will travel faster in water, and slower in air.
Answer : The correct option is, (b) Thermometer B, because it measures temperature more accurately than A.
Explanation :
Accuracy : It is defined as the closeness of a measured value to a standard or known value.
For Example: If the mass of a substance is 50 kg and one person weighed 48 kg and another person weighed 55 kg. Then, the weight measured by first person is more accurate.
Precision : It is defined as the closeness of two or more measurements to each other.
For Example: If you weigh a given substance five times and you get 1.8 kg each time. Then the measurement is said to be precise.
Level of precision is determined by the maximum number of decimal places.
As per question, the thermometer B has markings 40°, 45° and 50° at equal intervals and there are four small markings equally spaced in between the numbers and thermometer B has markings 40°, 45° and 50° at equal intervals but there is no small markings equally spaced in between the numbers. That means, thermometer B is measures temperature more accurately than thermometer A.
Hence, correct option is, (b) Thermometer B, because it measures temperature more accurately than A.
You can use them by analyzing the way you can solve it.