Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20
A particle with a charge of -1.24 10 C is moving with
instantaneous velocity 14.19 10 m/s) +
(-3.85 10 m/s). What is the force exerted on this
particle by a magnetic field (a) (1.40 T)
and (b) (1.40 T) ?
Short Answer
Expert verified
(a) ; (b) .
Step by step solution
01
Understand the Given Problem
We need to find the force exerted by a magnetic field on a charge with a specific velocity vector in two different scenarios. The charge is given as , and the velocity vector is . We need to find the force for two different magnetic fields: and .
02
Use the Lorentz Force Formula
The magnetic force on a charged particle moving with velocity in a magnetic field is given by the cross product . We'll need to compute this for each magnetic field scenario.
03
Calculate Cross Product for (a)
For , the cross product is . Since a vector crossed with itself is zero, and the cross product between vectors on the same axis is zero, this simplifies to: . Calculating gives: .
04
Calculate Force for (a)
Using and the cross product from Step 3, we find: . This gives: .
05
Calculate Cross Product for (b)
For , the cross product is .The result is calculated using the determinant method for cross products:. Which results in:.
06
Calculate Force for (b)
Using and the cross product from Step 5, we find:.This results in: .
Over 30 million students worldwide already upgrade their
learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Magnetic Field
A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. It's visualized as "lines of force" that exit through a magnet's north pole and enter through its south pole. Magnetic fields can be created by permanent magnets or flows of electric current. When a moving charge enters a magnetic field, it experiences a force. This force's direction depends on the charge's velocity and the magnetic field's direction. The strength of a magnetic field is measured in Tesla (T).
Key Points:
A magnetic field exerts a force on moving charges.
The magnetic field's direction and the charge’s motion will determine the direction of this force.
Strong magnetic fields exert stronger forces on charges.
In the context of the original exercise, recognizing which direction the magnetic fields ( and ) are acting in is crucial for calculating the resulting force on the charged particle.
Cross Product
The cross product is a vector multiplication operation that yields a vector perpendicular to two input vectors in three-dimensional space. This operation is central in physics, particularly in electromagnetism, where it is used to determine the force experienced by a charged particle within a magnetic field. Mathematically, the cross product of two vectors and is denoted as . The magnitude of the resulting vector is given by , where is the angle between the original vectors. Key Characteristics:
A cross product of parallel vectors is zero since .
The resulting vector is perpendicular to the original vectors.
It is used to calculate torques, rotations, and forces like the Lorentz force.
In the exercise, the cross product helps determine , which then allows calculation of the Lorentz force on the particle through .
Vector Calculus
Vector calculus is a mathematical tool used to study vector fields, which are functions assigning a vector to each point in space. It's vital in physics for understanding phenomena involving fields, such as electromagnetic fields. Vector operations, such as addition, scalar multiplication, and cross products, help solve complex physical problems.
Important Operations:
Vector Addition: Combines two vectors' magnitudes and directions.
Scalar Multiplication: Involves multiplying a vector by a scalar (number), scaling its magnitude.
Cross Product: Yields a vector perpendicular to two given vectors, especially crucial in calculating forces in electromagnetism.
In our task, vector calculus is applied to the velocity vector and the magnetic field to determine the force using the cross product. This application exemplifies the real-world utility of vector calculus in physics.
One App. One Place for Learning.
All the tools & learning materials you need for study success - in one app.
This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept
Privacy & Cookies Policy
Privacy Overview
This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the ...
Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.
Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.