Problem 1: A 3.0-cm-diameter (De), 10-turn coil of wire, located at z = 0 in the xy-plane, carries a current I = 2.5 A. A 2.0-mm-diameter (D₁) conducting loop with R = 2.0 × 10-4 resistance is also in the xy-plane at the cen- ter of the coil. At t = 0s, the loop begins to move along the z-axis with a constant speed of v = 75 m/s. What is the induced current in the conducting loop at t = 200 µs? 10 turns ·x ZA al FQ FIG. 1: The scheme for Problem 1 a) The magnetic field created by the coil looks in general as shown in Fig. 1, but since the diameter of the loop is much smaller than that of the coil, we can use the formula for the on-axis magnetic field of IR² the coil. For a single turn of the coil, this formula gives B = 2(2²+R²) 3/2, where Re is the radius of the coil. Derive the formula for the magnetic flux through the conducting loop as a function of the coordinate z. b) Derive the formula for the emf, & = |d, induced in the loop as it is moving up. To take the derivative of flux with respect to time, you need to use the chain rule for derivatives, d = dod. What is the meaning of d in this formula, and how does z depend on time? Express & as a function of t. dt c) Using the formula for & from the previous step and the resistance of the loop, compute the value of the induced current at t = 200 µs. (Answer: Iloop = 4.4 x 10-² A)
Problem 1: A 3.0-cm-diameter (De), 10-turn coil of wire, located at z = 0 in the xy-plane, carries a current I = 2.5 A. A 2.0-mm-diameter (D₁) conducting loop with R = 2.0 × 10-4 resistance is also in the xy-plane at the cen- ter of the coil. At t = 0s, the loop begins to move along the z-axis with a constant speed of v = 75 m/s. What is the induced current in the conducting loop at t = 200 µs? 10 turns ·x ZA al FQ FIG. 1: The scheme for Problem 1 a) The magnetic field created by the coil looks in general as shown in Fig. 1, but since the diameter of the loop is much smaller than that of the coil, we can use the formula for the on-axis magnetic field of IR² the coil. For a single turn of the coil, this formula gives B = 2(2²+R²) 3/2, where Re is the radius of the coil. Derive the formula for the magnetic flux through the conducting loop as a function of the coordinate z. b) Derive the formula for the emf, & = |d, induced in the loop as it is moving up. To take the derivative of flux with respect to time, you need to use the chain rule for derivatives, d = dod. What is the meaning of d in this formula, and how does z depend on time? Express & as a function of t. dt c) Using the formula for & from the previous step and the resistance of the loop, compute the value of the induced current at t = 200 µs. (Answer: Iloop = 4.4 x 10-² A)
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Chapter1: Units, Trigonometry. And Vectors
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Hello, I really need help with part A, part B and part C because I don't know how to do it and I keep getting the wrong answer, is there any chance you can help me with part A,part B and part C and can you label them as well
![### Problem 1:
A 3.0-cm-diameter (\(D_c\)), 10-turn coil of wire, located at \(z = 0\) in the \(xy\)-plane, carries a current \(I = 2.5 \, \text{A}\). A 2.0-mm-diameter (\(D_l\)) conducting loop with resistance \(R = 2.0 \times 10^{-4} \, \Omega\) is also in the \(xy\)-plane at the center of the coil. At \(t = 0 \, s\), the loop begins to move along the \(z\)-axis with a constant speed of \(v = 75 \, \text{m/s}\). What is the induced current in the conducting loop at \(t = 200 \, \mu \text{s}\)?
- **Fig. 1: The scheme for Problem 1** shows the coil with current \(I\), a magnetic field \(B\), and a loop moving along the \(z\)-axis.
### a)
The magnetic field created by the coil looks in general as shown in Fig. 1, but since the diameter of the loop is much smaller than that of the coil, we can use the formula for the on-axis magnetic field of the coil. For a single turn of the coil, this formula gives
\[
B = \frac{\mu_0}{2} \frac{IR_c^2}{(z^2 + R_c^2)^{3/2}}
\]
where \(R_c\) is the radius of the coil. Derive the formula for the magnetic flux \(\Phi\) through the conducting loop as a function of the coordinate \(z\).
### b)
Derive the formula for the emf, \(\mathcal{E} = \left| \frac{d\Phi}{dt} \right|\), induced in the loop as it is moving up. To take the derivative of flux with respect to time, you need to use the chain rule for derivatives,
\[
\frac{d\Phi}{dt} = \frac{d\Phi}{dz} \frac{dz}{dt}
\]
What is the meaning of \(\frac{dz}{dt}\) in this formula, and how does \(z\) depend on time? Express \(\mathcal{E}\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F63db6802-3a36-4dbd-9229-c565d17dfb62%2Fd25cc0dd-9f2d-4674-bcb2-1d261ffb768b%2F5l7qzi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 1:
A 3.0-cm-diameter (\(D_c\)), 10-turn coil of wire, located at \(z = 0\) in the \(xy\)-plane, carries a current \(I = 2.5 \, \text{A}\). A 2.0-mm-diameter (\(D_l\)) conducting loop with resistance \(R = 2.0 \times 10^{-4} \, \Omega\) is also in the \(xy\)-plane at the center of the coil. At \(t = 0 \, s\), the loop begins to move along the \(z\)-axis with a constant speed of \(v = 75 \, \text{m/s}\). What is the induced current in the conducting loop at \(t = 200 \, \mu \text{s}\)?
- **Fig. 1: The scheme for Problem 1** shows the coil with current \(I\), a magnetic field \(B\), and a loop moving along the \(z\)-axis.
### a)
The magnetic field created by the coil looks in general as shown in Fig. 1, but since the diameter of the loop is much smaller than that of the coil, we can use the formula for the on-axis magnetic field of the coil. For a single turn of the coil, this formula gives
\[
B = \frac{\mu_0}{2} \frac{IR_c^2}{(z^2 + R_c^2)^{3/2}}
\]
where \(R_c\) is the radius of the coil. Derive the formula for the magnetic flux \(\Phi\) through the conducting loop as a function of the coordinate \(z\).
### b)
Derive the formula for the emf, \(\mathcal{E} = \left| \frac{d\Phi}{dt} \right|\), induced in the loop as it is moving up. To take the derivative of flux with respect to time, you need to use the chain rule for derivatives,
\[
\frac{d\Phi}{dt} = \frac{d\Phi}{dz} \frac{dz}{dt}
\]
What is the meaning of \(\frac{dz}{dt}\) in this formula, and how does \(z\) depend on time? Express \(\mathcal{E}\
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