# Calculating Induced Current in a Solenoid-Loop Circuit

• geostudent
In summary, the conversation discussed a problem involving a cylindrical solenoid with a two-turn rectangular loop and a battery. The goal was to determine the magnitude and direction of the current in the loop one microsecond after it was connected to the battery. The conversation included equations for magnetic field, inductance, and current, and the general approach was deemed correct although there may have been some errors in the calculations.
geostudent

## Homework Statement

A cylindrical solenoid 30 cm long with a radius of 8 mm has 400 tightly-wound turns of wire uniformly distributed along its length (see the figure). Around the middle of the solenoid is a two-turn rectangular loop 3 cm by 2 cm made of resistive wire having a resistance of 190 ohms. One microsecond after connecting the loose wire to the battery to form a series circuit with the battery and a 20 resistor, what is the magnitude of the current in the rectangular loop and its direction (clockwise or counter-clockwise in the diagram)? (The battery has an emf of 9 V.)

## Homework Equations

B = μ$_{0}$NI /d

L= μ$_{0}$N$^{2}$$\pi$R$^{2}$/d

I = emf/R * [1-e$^{-(R/L)t}$]

emf(induced)= d$\Phi$/dt

## The Attempt at a Solution

I just want to check my reasoning here and get advice on how to approach a problem like this.

1. Since the Current is varying with time, I used I = emf/R * [1-e$^{-(R/L)t}$] to find I at t=1microsecond and got I= .20194 Amperes

2. Used B = μ$_{0}$NI /d to find B=1.76229E-4 Tesla

3. Induced Emf = -d$\Phi$/dt where $\Phi$ = ∫B*dA

I took dA to be the cross-sectional Area of the rectangle

Then induced (EMF*Number of turns in rectangle)/R =I

I keep getting relatively close answers but not correct. I don't think I'm thinking of this n the right way. Where am I at fault?

The general approach looks alright, although I'm not sure what you're doing with the value of the current in the coil or the value of the B field for that particular time. It'll be the rate of change of the B field that you'll need, no?

Even so, the value you're getting for the current looks a bit odd. What values did you calculate for the inductance and the time constant?

## What is induced current in a coil?

Induced current in a coil is the flow of electric charge that is generated in a coil of wire when the magnetic field through the coil changes. This change in magnetic field can be created by moving the coil in a stationary magnetic field, changing the strength of the magnetic field, or changing the orientation of the coil with respect to the magnetic field.

## How is induced current different from direct current?

Induced current is different from direct current in that it is not produced by a battery or other external source of electric charge. Instead, it is created by the changing magnetic field passing through the coil, which induces a voltage that causes the current to flow.

## What is Faraday's law of induction?

Faraday's law of induction states that the induced electromotive force in a closed circuit is proportional to the rate of change of the magnetic flux through the circuit. In other words, the faster the magnetic field changes, the greater the induced current in the coil will be.

## What factors affect the strength of induced current in a coil?

The strength of induced current in a coil is affected by several factors, including the strength of the magnetic field, the number of turns in the coil, the speed at which the coil moves through the magnetic field, and the resistance of the coil.

## What are some practical applications of induced current in a coil?

Induced current in a coil has many practical applications, such as in generators, transformers, and electric motors. It is also used in devices such as metal detectors, induction cooktops, and wireless charging systems.

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