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Chapter 19: Problem 1

A proton is projected into a magnetic field that is directed along the positive \(x\) -axis. Find the direction of the magnetic force exerted on the proton for each of the following directions of the proton's velocity: (a) the positive \(y\) -direction; (b) the negative \(y\) -direction; (c) the positive \(x\) -direction.

Short Answer

Expert verified
The directions of the magnetic force on the proton for velocity directions (a) positive y, (b) negative y, and (c) positive x are respectively: (a) negative z-direction, (b) positive z-direction, and (c) no force or zero.

Step by step solution

01

Apply the right-hand rule for positive y-direction

Applying the right-hand rule, align the hand with the direction of the velocity of the proton (thump points in the positive y-direction), then curl the fingers in the direction of the magnetic field (Results in fingers pointing in the x-direction), the extended palm will give the direction of the force which is in the negative z-direction.
02

Apply the right-hand rule for negative y-direction

For the negative y-direction of the proton's velocity, align the hand with the direction of the velocity (thumb points in the negative y-direction), curl the fingers to the direction of the field (x-direction). The force ends up pointing in the positive z-direction.
03

Apply the right-hand rule for positive x-direction

For the proton moving in the same direction as the magnetic field, i.e., the positive x-direction, there is no force (thumb and fingers fall in the same line). Therefore, the force exerted on the proton is zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Right-Hand Rule - Understanding Its Fundamentals
The right-hand rule is a practical tool used to determine the direction of certain vectors in physics, such as the magnetic force on a moving charge. When dealing with magnetic fields and forces, the right-hand rule helps us ascertain the force direction based on velocity and magnetic field direction. Here's a straightforward way to use it:
  • Extend your right hand, so the thumb is perpendicular to the fingers.
  • Have your thumb represent the direction of the positive charge's velocity.
  • Let your fingers represent the direction of the magnetic field.
  • Your palm will then face in the direction of the magnetic force exerted on that charge.

For positively charged particles like protons, this rule is immensely helpful in visualizing how magnetic forces interact in three-dimensional space. Remember, physics is all about following these rules to decode nature's pattern.
Proton Velocity - Its Critical Role in Magnetic Force
The velocity of a proton plays a pivotal role in determining the direction of the magnetic force exerted upon it. Since protons are positively charged particles, their movement through a magnetic field results in experiencing Lorentz force, which is perpendicular to both the velocity and the magnetic field. In other words, the way a proton moves significantly alters how it interacts with the magnetic field.
  • When a proton moves along the positive y-direction, for instance, the magnetic force is directed perpendicularly due to the cross product of velocity and field vectors.
  • Change the direction of the velocity and the direction of magnetic force changes accordingly, as with the negative y-direction or along the x-axis.
  • The right-hand rule again comes in handy to visualize the path of the force depending on velocity direction.

Understanding these dynamics helps explain many phenomena, such as the movement of charged particles in Earth's magnetic field. Remember, the velocity of the charged particle is an essential ingredient in determining magnetic interactions.
Magnetic Field Direction and Its Effect
The magnetic field direction is another fundamental aspect impacting the force experienced by a proton moving through it. Typically, this direction is visualized as lines going from the magnetic north to the south pole, though in exercises like ours, it's often described with specific components along the coordinate axes. Here's why it's important:
  • A magnetic field directed along the x-axis, for instance, simplifies analysis because it gives one clear component to consider.
  • Whenever particles like protons enter such fields, the force's direction becomes quick and direct to determine, using consistent geometry powered by vector mathematics.
  • Any perpendicular charge velocity will result in magnetic force action, whereas parallel velocity holds no force due to the nature of vector projections 鈥� this is why a proton with velocity in the x-direction didn't experience a force.

Comprehending the alignment and effect of magnetic fields provides deeper insights into not just textbook problems but also real-world electromagnetic applications.

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Most popular questions from this chapter

A proton is at rest at the plane vertical boundary of a region containing a uniform vertical magnetic field \(B\). An alpha particle moving horizontally makes a head-on elastic collision with the proton. Immediately after the collision, both particles enter the magnetic field, moving perpendicular to the direction of the field. The radius of the proton's trajectory is \(R\). Find the radius of the alpha particle's trajectory. The mass of the alpha particle is four times that of the proton, and its charge is twice that of the proton.

A laboratory electromagnet produces a magnetic field of magnitude \(1.50 \mathrm{~T}\). A proton moves through this field with a speed of \(6.00 \times 10^{6 \mathrm{~m} / \mathrm{s} . \text { (a) Find the mag- }}\) nitude of the maximum magnetic force that could be exerted on the proton. (b) What is the magnitude of the maximum acceleration of the proton? (c) Would the field exert the same magnetic force on an electron moving through the field with the same speed? Would the electron undergo the same acceleration? Explain.

A wire carries a current of \(10.0 \mathrm{~A}\) in a direction that makes an angle of \(30.0^{\circ}\) with the direction of a magnetic field of strength \(0.300 \mathrm{~T}\). Find the magnetic force on a \(5.00-\mathrm{m}\) length of the wire.

A \(0.200-\mathrm{kg}\) metal rod carrying a current of \(10.0 \mathrm{~A}\) glides on two horizontal rails \(0.500 \mathrm{~m}\) apart. What vertical mag netic field is required to keep the rod moving at a constant speed if the coefficient of kinetic friction between the rod and rails is \(0.100\) ?

At a certain location, Earth has a magnetic field of \(0.60 \times 10^{-4} \mathrm{~T}\), pointing \(75^{\circ}\) below the horizontal in a north-south plane. A \(10.0\) -m-long straight wire carries a \(15-A\) current. (a) If the current is directed horizontally toward the east, what are the magnitude and direction of the magnetic force on the wire? (b) What are the magnitude and direction of the force if the current is directed vertically upward?
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