The main magnet in an MRI machine is a super-conducting solenoid 1.8 m long and 30 cm in radius.

step 1 High in the given problem length of the solenoid is l is equal to 1 .8 meter. Its radius is 30 c

The main magnet in an MRI machine is a super-conducting...

The main magnet in an MRI machine is a super-conducting solenoid $1.8 \mathrm{m}$ long and $30 \mathrm{cm}$ in radius. During normal operation, the current through the windings is 100 A, the resistance of the windings is zero, and the magnetic field magnitude is $1.5 \mathrm{T}$. (a) What is its inductance? (b) The magnet is started by connecting the solenoid to a power supply. It takes $8.0 \mathrm{min}$ for the current to go from zero to $100 \mathrm{A}$ What is the emf of the power supply?

The main magnet in an MRI machine is a super-conducting solenoid $1.8 \mathrm{m}$ long and $30 \mathrm{cm}$ in radius. During normal operation, the current through the windings is 100 A, the resistance of the windings is zero, and the magnetic field magnitude is $1.5 \mathrm{T}$.
(a) What is its inductance?
(b) The magnet is started by connecting the solenoid to a power supply. It takes $8.0 \mathrm{min}$ for the current to go from zero to $100 \mathrm{A}$ What is the emf of the power supply?

Key Concepts

Solenoid

A solenoid is a cylindrical coil of wire that produces a nearly uniform magnetic field when a current passes through it. In many practical applications, such as MRI machines, solenoids are used due to their ability to generate strong and consistent magnetic fields. The design and properties of a solenoid, including its length, radius, and number of turns, directly influence its inductance and magnetic field distribution.

Rate of Change of Current

The rate at which current changes over time (dI/dt) plays a critical role in determining the magnitude of the induced emf in an inductor. A slow change in current leads to a small induced voltage, while a rapid change produces a larger emf. This relationship is fundamental in designing circuits that involve ramping up or down currents.

Inductance

Inductance quantifies the ability of a coil or solenoid to store magnetic energy when a current flows through it. It is a fundamental property in electromagnetism, defined as the ratio of the magnetic flux linkage to the current. Inductance is crucial in circuits with time-varying currents because it determines how much an induced voltage (emf) will oppose changes in current.

Self-induced emf

When the current through an inductor changes, a self-induced electromotive force (emf) is generated according to Faraday's law of induction. This induced emf, expressed as the product of the inductance and the rate of change of the current (V = L dI/dt), always opposes the change in current, a principle known as Lenz's law. This concept is key when analyzing transient behavior in circuits with inductors.

Transcript

Please give Ace some feedback