At t = 0, a battery is connected to a series arrangement of a resistor and an inductor. If the inductive time constant is 36.0 ms, at what time is the rate at which energy is dissipated in the resistor equal to the rate at which energy is stored in the inductor's magnetic field?

The rate at which energy is dissipated in the resistor is equal to the rate at which energy is stored in the inductor's magnetic field in 24.95 ms.

Explanation:

The energy stored in the inductor is given as

E₁ = ½LI²

The rate at which energy is stored in the inductor is

(dE₁/dt) = (d/dt) (½LI²)

Since L is a constant

(dE₁/dt) = ½L × 2I (dI/dt) = LI (dI/dt)

(dE₁/dt) = LI (dI/dt)

Rate of Energy dissipated in a resistor = Power = I²R

(dE₂/dt) = I²R

When the rate at which energy is dissipated in the resistor equal to the rate at which energy is stored in the inductor's magnetic field

(dE₁/dt) = (dE₂/dt)

OK (dI/dt) = I²R

L (dI/dt) = IR

Current in a this kind of series setup of inductor and resistor at any time, t, is given as

I = (V/R) (1 - e⁻ᵏᵗ)

k = (1/time constant) = (R/L)

(dI/dt) = (kV/R) e⁻ᵏᵗ = (RV/RL) e⁻ᵏᵗ = (V/L) e⁻ᵏᵗ

L (dI/dt) = IR

L [(V/L) e⁻ᵏᵗ] = R [(V/R) (1 - e⁻ᵏᵗ)

V e⁻ᵏᵗ = V (1 - e⁻ᵏᵗ)

e⁻ᵏᵗ = 1 - e⁻ᵏᵗ

2 e⁻ᵏᵗ = 1

e⁻ᵏᵗ = (1/2) = 0.5

e⁻ᵏᵗ = 0.5

In e⁻ᵏᵗ = In 0.5 = -0.69315

- kt = -0.69315

kt = 0.69315

k = (1/time constant)

Time constant = 36.0 ms = 0.036 s

k = (1/0.036) = 27.78

27.78t = 0.69315

t = (0.69315/27.78) = 0.02495 = 24.95 ms

Hope this Helps!!!

explanation:

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carbon is found in all living things. carbon atoms move constantly through

living organisms, the oceans, the atmosphere, and the earth’s crust in what is

known as the carbon cycle. the directions taken by carbon atoms through this

cycle are very complicated and can take millions of years to make a full circle.

all animals, from humans to the dinosaurs are part of the carbon cycle. when

animals eat food, they get carbon in the form of carbohydrates and proteins.

in animals, oxygen combines with food in the cells to produce energy for daily

activity and then gives off carbon. the carbon combines with oxygen to form

carbon dioxide (co2) and is released back into the atmosphere as a waste

product when animals breathe and exhale.

from 145-65 million years ago, earth was much

hotter than today and covered with dense,

tropical swamp forests. the trees and other

plants were immense and

provided an endless supply of

food for the giant animals that

roamed the land. somewhere in

the air above one of these

forests, a lone carbon atom has joined

up with two oxygen atoms to form a molecule of co2.

the co2 molecule was sucked into the tiny holes (stomata) on the leaf of a

fern plant and joined with sunlight, chlorophyll and water to make food and

energy in the plant’s cells through photosynthesis. the oxygen (o2) from the

co2 molecule was sent back into the atmosphere; the carbon atom (c) was

detached and used to make a molecule of sugar.

if that carbon atom had been eaten instead, a totally

different situation would have developed. let’s say

that a giant plant-eating dinosaur (herbivore) ate the

fern for breakfast, and swallowed the carbon in the

fern in the form of carbohydrates and proteins. in

the process of respiration, oxygen combined with

the carbohydrates and proteins in the dinosaur’s cells

to provide energy for its daily activity.