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Problem 3: A flat, circular disk of radius R is uniformly charged with to- tal charge Q. The disk spins at angular velocity @ about an axis through its center (see Fig.3). What is the magnetic field strength at the center of the disk? dr b) Find the magnetic field dB center created by this ring at the center of the disk in terms of Q, R, w, dr, and other relevant constants. wat R a) Choose a ring of width dr and radius r inside the disk, as shown in Fig.3. The amount of charge dq that passes through a cross-section of this ring in the interval of time dt is enclosed in the hatched section of this ring. Compute dq from the surface charge density of the disk and the area of the hatched region (note that the length of an arc of a circle with radius r is equal to r0, where is the angle in radians which the arc subtends at the center of the circle; see the scheme in Fig.4). Compute the current I flowing through this thin ring as dq/dt. FIG. 3: The scheme for Problem 3 3 |гө 2 FIG. 4: Arc length c) Sum up the contributions from all the rings by taking the integral Bcenter = ₁ dBcenter (what are the limits of integration?). Answer: Bcenter HoQw 2лR

Problem 3: A flat, circular disk of radius R is uniformly charged with to- tal charge Q. The disk spins at angular velocity @ about an axis through its center (see Fig.3). What is the magnetic field strength at the center of the disk? dr b) Find the magnetic field dB center created by this ring at the center of the disk in terms of Q, R, w, dr, and other relevant constants. wat R a) Choose a ring of width dr and radius r inside the disk, as shown in Fig.3. The amount of charge dq that passes through a cross-section of this ring in the interval of time dt is enclosed in the hatched section of this ring. Compute dq from the surface charge density of the disk and the area of the hatched region (note that the length of an arc of a circle with radius r is equal to r0, where is the angle in radians which the arc subtends at the center of the circle; see the scheme in Fig.4). Compute the current I flowing through this thin ring as dq/dt. FIG. 3: The scheme for Problem 3 3 |гө 2 FIG. 4: Arc length c) Sum up the contributions from all the rings by taking the integral Bcenter = ₁ dBcenter (what are the limits of integration?). Answer: Bcenter HoQw 2лR

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Hello,  I really need help with part A, Part B and Part C. I was wondering if you can help me with the problems because I don't understand it and I was wondering if you can label which one is so I can following along with the problems. thank you

**Problem 3:** A flat, circular disk of radius \( R \) is uniformly charged with total charge \( Q \). The disk spins at angular velocity \( \omega \) about an axis through its center (see Fig. 3). What is the magnetic field strength at the center of the disk? **a)** Choose a ring of width \( dr \) and radius \( r \) inside the disk, as shown in Fig. 3. The amount of charge \( dq \) that passes through a cross-section of this ring in the interval of time \( dt \) is enclosed in the hatched section of this ring. Compute \( dq \) from the surface charge density of the disk and the area of the hatched region (note that the length of an arc of a circle with radius \( r \) is equal to \( r\theta \), where \( \theta \) is the angle in radians which the arc subtends at the center of the circle; see the scheme in Fig. 4). Compute the current \( I \) flowing through this thin ring as \( dq/dt \). **b)** Find the magnetic field \( dB_{\text{center}} \) created by this ring at the center of the disk in terms of \( Q, R, \omega, dr \), and other relevant constants. **c)** Sum up the contributions from all the rings by taking the integral \( B_{\text{center}} = \int_r dB_{\text{center}} \) (what are the limits of integration?). **Answer:** \( B_{\text{center}} = \frac{\mu_0 Q \omega}{2 \pi R} \). **Figures:** - **Fig. 3**: Displays a diagram of the disk with a highlighted ring of radius \( r \) and width \( dr \). The ring is moving with angular velocity \( \omega \) and encloses a hatched area that represents the charge \( dq \) moving through the ring over time \( dt \). - **Fig. 4**: Illustrates the concept of arc length, showing a circle segment with radius \( r \) subtending an angle \( \theta \). This exercise helps in understanding how the motion of charged particles generates magnetic fields and explores integrating contributions from multiple elements to find total effects in physics.
Transcribed Image Text:**Problem 3:** A flat, circular disk of radius \( R \) is uniformly charged with total charge \( Q \). The disk spins at angular velocity \( \omega \) about an axis through its center (see Fig. 3). What is the magnetic field strength at the center of the disk? **a)** Choose a ring of width \( dr \) and radius \( r \) inside the disk, as shown in Fig. 3. The amount of charge \( dq \) that passes through a cross-section of this ring in the interval of time \( dt \) is enclosed in the hatched section of this ring. Compute \( dq \) from the surface charge density of the disk and the area of the hatched region (note that the length of an arc of a circle with radius \( r \) is equal to \( r\theta \), where \( \theta \) is the angle in radians which the arc subtends at the center of the circle; see the scheme in Fig. 4). Compute the current \( I \) flowing through this thin ring as \( dq/dt \). **b)** Find the magnetic field \( dB_{\text{center}} \) created by this ring at the center of the disk in terms of \( Q, R, \omega, dr \), and other relevant constants. **c)** Sum up the contributions from all the rings by taking the integral \( B_{\text{center}} = \int_r dB_{\text{center}} \) (what are the limits of integration?). **Answer:** \( B_{\text{center}} = \frac{\mu_0 Q \omega}{2 \pi R} \). **Figures:** - **Fig. 3**: Displays a diagram of the disk with a highlighted ring of radius \( r \) and width \( dr \). The ring is moving with angular velocity \( \omega \) and encloses a hatched area that represents the charge \( dq \) moving through the ring over time \( dt \). - **Fig. 4**: Illustrates the concept of arc length, showing a circle segment with radius \( r \) subtending an angle \( \theta \). This exercise helps in understanding how the motion of charged particles generates magnetic fields and explores integrating contributions from multiple elements to find total effects in physics.
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