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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 dBcenter 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 re, where is the angle in radians which the are 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 r re FIG. 4: Arc length c) Sum up the contributions from all the rings by taking the integral Bcenter = f, dBcenter (what are the HoQw limits of integration?). Answer: Bcenter = 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 dBcenter 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 re, where is the angle in radians which the are 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 r re FIG. 4: Arc length c) Sum up the contributions from all the rings by taking the integral Bcenter = f, dBcenter (what are the HoQw limits of integration?). Answer: Bcenter = 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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I need help with part A, part B and part C I was wondering if u help me with them and can you label which one is which 

**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 \( dq/dt \). Figure 3: The scheme for Problem 3 shows a circular disk with markings depicting angular velocity, the ring selected with width \( dr \), and defined radius \( r \). It also indicates the angular displacement \( \omega dt \) for a short time interval. **b)** Find the magnetic field \( dB_{center} \) created by this ring at the center of the disk in terms of \( Q, R, \omega, dr \), and other relevant constants. Figure 4: A diagram labeled "Arc length" illustrates a section of the circle with radius \( r \) and angle \( \theta \). **c)** Sum up the contributions from all the rings by taking the integral \( B_{center} = \int dB_{center} \) (what are the limits of integration?). Answer: \( B_{center} = \frac{\mu_0 Q \omega}{2 \pi R} \).
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 \( dq/dt \). Figure 3: The scheme for Problem 3 shows a circular disk with markings depicting angular velocity, the ring selected with width \( dr \), and defined radius \( r \). It also indicates the angular displacement \( \omega dt \) for a short time interval. **b)** Find the magnetic field \( dB_{center} \) created by this ring at the center of the disk in terms of \( Q, R, \omega, dr \), and other relevant constants. Figure 4: A diagram labeled "Arc length" illustrates a section of the circle with radius \( r \) and angle \( \theta \). **c)** Sum up the contributions from all the rings by taking the integral \( B_{center} = \int dB_{center} \) (what are the limits of integration?). Answer: \( B_{center} = \frac{\mu_0 Q \omega}{2 \pi R} \).
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