A coil with an inductance of 2.5 H and a resistance of 15 Ω is suddenly connected to an ideal battery with ε = 140 V. At 0.15 s after the connection is made, what is the rate at which (a) energy is being stored in the magnetic field, (b) thermal energy is appearing in the resistance, and (c) energy is being delivered by the battery?

Explanation:

Given that,

Inductance

L = 2.5H

Resistance

R = 15 Ω

EMF of battery

ε = 140 V

Time

t = 0.15s

A. Rate at which Energy stored in the magnetic field

Energy stored can be determined using

U = ½Li²

Then, the rate is

dU/dt = d(½Li²)/dt

Then inductance is constant

Then, dU/dt = Li• di/dt

Where i(t) is give as

i(t) = ε/R (1 —exp(-t/τ))

Where τ is time constant and it is given as

τ = L/R = 2.5/15 = 0.167 seconds

Getting the current I(t)

i(t) = ε/R (1 —exp(-t/τ))

I(t) = 140/15(1—exp(-0.15/0.167)

I(t) = 9.33 × (1—0.407)

I(t) = 9.33 × 0.593

I(t) = 5.54 A

Getting di/dt

di/dt = ε/R (- -1/τ•exp(-t/τ)

di/dt = ε/Rτ exp(-t/τ)

Substituting the given values

di/dt = 140/(15×0.167) exp(-0.15/0.167)

di/dt = 56 × exp(-0.9)

di/dt = 22.78 A/s

Then,

dU/dt = Li• di/dt

dU/dt = 2.5 × 5.54 × 22.78

dU/dt = 315.32 Watts

B. Rate of thermal energy appearing in the resistance

The power dissipated in the resistor can be determined using

P = i²R

We already know the current at t = 0.15s

I = 5.54A and R = 15 ohms

Then,

P = i²R

P = 5.54² × 15

P = 460.37 Watts

C. Rate of energy delivered by the battery

P = iV

P = 5.54 × 140

P = 775.6 Watts

2b2t

explanation:

2b2t

answer: at 24 february, 2014 the population gets twice of its initial population.

explanation:

since we have given that

the population for alpha city, t years after january 1, 2004 is given as

first we find out the initial population, i.e. t=0,

so, our quadratic equation becomes,

according to question, we have said that if the population has twice its initial population, then it becomes

so, time taken to reach this above mentioned population is given by

now, we know that time can't be negative so, we take t=9.149 years.

hence, at 24 february, 2014 the population gets twice of its initial population.