Chapter 7: Problem 6
A disk of radius
Short Answer
Expert verified
Answer: The magnetic field at any point on the -axis due to the rotating charged disk is given by the expression: .
Step by step solution
01
Use the Biot-Savart law to find magnetic field due to rotating disk
First, let's find the magnetic field produced by the rotating disk. To do this, we will use the Biot-Savart law:
where is the velocity of the charge at the element , and is the position of the element, is the position where we want to find the magnetic field, and is the vector pointing from the element to the observation point. The integral is taken over the whole disk.
02
Simplify the expression
We are interested in finding the magnetic field on the axis. So, let's choose the observation point . The position of the charged element on the disk . We can express the velocity of the charged element in polar coordinates:
Now, let's replace this expression for into the Biot-Savart law expression:
We now need to convert this integral into polar coordinates:
Where and are the polar coordinates of the charged element, and and are the unit vectors in the polar coordinate system.
03
Simplify the integral
Now we substitute the polar coordinates expressions, and the Biot-Savart law becomes:
Notice that due to symmetry, the -component and -component of will cancel each other out. Also, (since we're not integrating along the direction), and thus only need to consider the -component of .
Now, perform the cross product and integrate:
After integrating with respect to ,
Now, perform the remaining integral with respect to :
Finally, we get the expression for the magnetic field on the -axis:
This is the magnetic field at any point on the -axis due to the rotating charged disk.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Magnetic Field
The magnetic field, often denoted as or , is a fundamental concept in physics that describes how magnetic forces are distributed in space. In this problem, we are looking at the magnetic field created by a rotating disk of charge. Utilizing the Biot-Savart Law, which helps us calculate the magnetic field generated by a current, is crucial.
To apply Biot-Savart Law in this scenario, consider that the movement of the charged particles as the disk rotates is equivalent to an electric current. This current is what generates the magnetic field .
The expression for the magnetic field along the -axis due to the disk is derived from integrating all contributions to the magnetic field (from each small piece of the disk) and considering symmetry properties. It makes it more straightforward since only the -component of the magnetic field remains after accounting for symmetry.
To apply Biot-Savart Law in this scenario, consider that the movement of the charged particles as the disk rotates is equivalent to an electric current. This current is what generates the magnetic field
The expression for the magnetic field
Surface Charge Density
Surface charge density, represented as , is an important quantity that describes how much electric charge is distributed over a surface area. It's measured in coulombs per square meter (C/m²). In this exercise, the disk possesses a uniform surface charge density, which means the charge is spread evenly over its entire surface.
The surface charge density impacts the magnitude of the magnetic field generated by the rotating disk. Higher means more charge is involved in the rotation, and therefore, the resulting magnetic field will be stronger. In the formula for , directly influences the overall expression through multiplication, demonstrating its role in scaling the magnetic field intensity.
The surface charge density impacts the magnitude of the magnetic field generated by the rotating disk. Higher
Angular Velocity
Angular velocity, denoted by , describes how fast an object rotates about an axis. It is measured in radians per second (rad/s). For a rotating body like the disk in this problem, the angular velocity determines how quickly the disk is spinning.
The angular velocity is critical in calculating the magnetic field because it dictates the speed at which the charged particles are moving. This speed translates into an equivalent electric current, which is fundamental in generating the magnetic field via the Biot-Savart Law. Thus, appears in the solution formula for the magnetic field, highlighting its direct effect on the magnitude of .
The angular velocity is critical in calculating the magnetic field because it dictates the speed at which the charged particles are moving. This speed translates into an equivalent electric current, which is fundamental in generating the magnetic field via the Biot-Savart Law. Thus,
Polar Coordinates
Polar coordinates offer a convenient way to describe positions on a plane using a distance from a reference point (origin) and an angle from a reference direction. In this exercise, polar coordinates are used to represent the positions of infinitesimal charge elements on the disk.
Switching to polar coordinates simplifies integrating the contributions to the magnetic field, especially when considering rotational symmetry. Expressing and in polar form as and allows transforming the integral into a form that's more manageable and reflective of the disk's geometric nature.
Overall, polar coordinates help streamline the solution process by taking advantage of symmetry and making mathematical expressions more tractable.
Switching to polar coordinates simplifies integrating the contributions to the magnetic field, especially when considering rotational symmetry. Expressing
Overall, polar coordinates help streamline the solution process by taking advantage of symmetry and making mathematical expressions more tractable.