A deuteron (the nucleus of an isotope of hydrogen) has a mass of 3.34 ×10^-27 kg and a charge of

step 1 So this problem is problem number 27 .21. Okay. So we have a deuterine of mass 3 .34 times 10 to

A deuteron (the nucleus of an isotope of hydrogen) has a...

A deuteron (the nucleus of an isotope of hydrogen) has a mass of 3.34 $\times$ 10$^{-27}$ kg and a charge of $+e$. The deuteron travels in a circular path with a radius of 6.96 mm in a magnetic field with magnitude 2.50 T. (a) Find the speed of the deuteron. (b) Find the time required for it to make half a revolution. (c) Through what potential difference would the deuteron have to be accelerated to acquire this speed?

A deuteron (the nucleus of an isotope of hydrogen) has a mass of 3.34 $\times$ 10$^{-27}$ kg and a charge of $+e$. The deuteron travels in a circular path with a radius of 6.96 mm in a magnetic field with
magnitude 2.50 T. (a) Find the speed of the deuteron. (b) Find the time required for it to make half a revolution. (c) Through what potential difference would the deuteron have to be accelerated to acquire this speed?

Key Concepts

Lorentz Force

The Lorentz force is the fundamental force acting on a charged particle moving in a magnetic field. When the particle’s velocity is perpendicular to the magnetic field, this force has a magnitude equal to the product of the charge, the speed of the particle, and the magnetic field strength, and it always acts perpendicular to the motion. This causes the particle to move in a circular path, with the magnetic force providing the necessary centripetal force for this motion.

Uniform Circular Motion

Uniform circular motion describes the motion of an object traveling at a constant speed along a circular path. Key parameters include the radius of the circle, the period (the time for one complete revolution), and the relationship between the centripetal force and the mass and velocity of the particle. In scenarios involving charged particles in magnetic fields, the centripetal force is provided by the magnetic Lorentz force.

Energy Conversion and Particle Acceleration

In processes where charged particles are accelerated through a potential difference, the electrical potential energy is converted into kinetic energy. The relationship is given by equating the gain in kinetic energy to the product of the charge and the potential difference. This concept is essential for understanding how particles achieve sufficient speeds before entering a magnetic field, thereby linking the accelerator’s potential difference to the speed of the particle.

Transcript

00:01 So this problem is problem number 27 .21.

00:04 Okay.

00:06 So we have a deuterine of mass 3 .34 times 10 to the negative 27 kilograms.

00:18 So i'm going to write down here mass of deuterine.

00:22 That is 3 .34 times 10 to the negative 27 kilogram.

00:31 Okay.

00:34 Let me write down the problem number.

00:39 Here 27 .21.

00:48 So mass of the deuterine is 3 .3, 4 times 10 to the negative 27 kilogram.

00:54 And this is moving in a magnetic field, magnetic field of the magnitude 2 .5 tesla.

01:03 Now, when a charged particle moves in magnetic field, it moves in a circle.

01:08 In this case, we have the radius of the circle is given, which is r equals 6 .96 millimeter.

01:17 So i'm just going to write down 6 .96 times 10 to negative 3 meter.

01:22 Ok.

01:25 So from this information we have here, we first want to find the speed of the due turn.

01:31 Ok.

01:33 So v equals what? so when a charged particle in magnetic field, when the charge particle moves in magnetic field, then the force is provided by the magnetic field, right? so the force of the magnetic field, which is b -e -v, has to be providing the necessary centripetal force, which is m -d -v -squared over r.

02:07 Okay, radius of the circle.

02:10 So from this information here, if you cancel out this to here, you will get v, equals b -e -r over md.

02:28 All right.

02:30 So this, i'm gonna write this down here.

02:33 This value is gonna be, we know the value of b, and e actually, i forgot to mention here, it actually is always, e has to be 1 .6 times 10 to the negative 19 column.

02:49 It's always that...

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