A varying current i(t) = 1(16 - 1) A (t in seconds) flows through a long straight wire that lies along the x-axis. The current; produces a magnetic field B whose magnitude at a distance r from the wire is B = T. Furthermore, at the point P, B points away from the observer as shown in the figure. 2xr Wire loop C Rectangular region R Volt meter Φ(t) = [E. x=0 B E. dr = P= (x, y) Calculate the flux (1), at time t, of B through a rectangle of dimensions L x H = 9 x 2 m whose top and bottom edges are parallel to the wire and whose bottom edge is located d = 0.5 m above the wire. Assume that the rectangle and the wire are located in the same plane. (Use symbolic notation and fractions where needed. Let I = i(t) and express your answer in terms of μ and I.) y x=L Use Faraday's Law to determine the voltage drop around the rectangular loop (the boundary of the rectangle) at time t = 6. Assume μ = 4x 10-7 T.m/A. (Use symbolic notation and fractions where needed.) T.m²
A varying current i(t) = 1(16 - 1) A (t in seconds) flows through a long straight wire that lies along the x-axis. The current; produces a magnetic field B whose magnitude at a distance r from the wire is B = T. Furthermore, at the point P, B points away from the observer as shown in the figure. 2xr Wire loop C Rectangular region R Volt meter Φ(t) = [E. x=0 B E. dr = P= (x, y) Calculate the flux (1), at time t, of B through a rectangle of dimensions L x H = 9 x 2 m whose top and bottom edges are parallel to the wire and whose bottom edge is located d = 0.5 m above the wire. Assume that the rectangle and the wire are located in the same plane. (Use symbolic notation and fractions where needed. Let I = i(t) and express your answer in terms of μ and I.) y x=L Use Faraday's Law to determine the voltage drop around the rectangular loop (the boundary of the rectangle) at time t = 6. Assume μ = 4x 10-7 T.m/A. (Use symbolic notation and fractions where needed.) T.m²
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![### Educational Content on Magnetic Flux and Voltage Drop
**Introduction:**
A varying current \( i(t) = t(16 - t) \) A, where \( t \) is in seconds, flows through a long straight wire along the x-axis. This current produces a magnetic field \( B \) with a magnitude at a distance \( r \) from the wire given by \( B = \frac{\mu_0 I}{2\pi r} \) T. At a specific point \( P \), the magnetic field \( B \) points away from the observer, as illustrated.
**Diagram Explanation:**
The illustration presents a wire loop \( C \) and a rectangular region \( R \), with a voltmeter connected. The rectangle has dimensions \( L \times H = 9 \times 2 \) meters, with top and bottom edges parallel to the wire. The rectangle’s bottom edge is located \( d = 0.5 \) meters above the wire. Both the rectangle and the wire are assumed to be in the same plane.
**Task:**
1. Calculate the magnetic flux \( \Phi(t) \) through the rectangle at time \( t \), using symbolic notation. Let \( I = i(t) \) and express the answer in terms of \( \mu_0 \) and \( I \).
\[
\Phi(t) = \_\_\_\_
\]
(Provide the answer in \(\text{T} \cdot \text{m}^2\))
2. Use Faraday’s Law to determine the voltage drop around the rectangular loop at time \( t = 6 \).
Assume \( \mu_0 = 4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A} \).
\[
\oint_{C} \mathbf{E} \cdot d\mathbf{r} = \_\_\_\_
\]
(Answer in volts, using symbolic notation and fractions where needed)
**Conclusion:**
By applying the given formulas and laws such as Faraday’s Law and properties of magnetic fields, one can determine both the magnetic flux through a rectangle in a magnetic field and the voltage drop around a loop at a specific moment in time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77548912-c51c-4c9d-8b51-f3905a3bec75%2F011f8390-2b37-4105-af08-a4788e331b3f%2Ffnyfb7h_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Content on Magnetic Flux and Voltage Drop
**Introduction:**
A varying current \( i(t) = t(16 - t) \) A, where \( t \) is in seconds, flows through a long straight wire along the x-axis. This current produces a magnetic field \( B \) with a magnitude at a distance \( r \) from the wire given by \( B = \frac{\mu_0 I}{2\pi r} \) T. At a specific point \( P \), the magnetic field \( B \) points away from the observer, as illustrated.
**Diagram Explanation:**
The illustration presents a wire loop \( C \) and a rectangular region \( R \), with a voltmeter connected. The rectangle has dimensions \( L \times H = 9 \times 2 \) meters, with top and bottom edges parallel to the wire. The rectangle’s bottom edge is located \( d = 0.5 \) meters above the wire. Both the rectangle and the wire are assumed to be in the same plane.
**Task:**
1. Calculate the magnetic flux \( \Phi(t) \) through the rectangle at time \( t \), using symbolic notation. Let \( I = i(t) \) and express the answer in terms of \( \mu_0 \) and \( I \).
\[
\Phi(t) = \_\_\_\_
\]
(Provide the answer in \(\text{T} \cdot \text{m}^2\))
2. Use Faraday’s Law to determine the voltage drop around the rectangular loop at time \( t = 6 \).
Assume \( \mu_0 = 4\pi \times 10^{-7} \, \text{T} \cdot \text{m/A} \).
\[
\oint_{C} \mathbf{E} \cdot d\mathbf{r} = \_\_\_\_
\]
(Answer in volts, using symbolic notation and fractions where needed)
**Conclusion:**
By applying the given formulas and laws such as Faraday’s Law and properties of magnetic fields, one can determine both the magnetic flux through a rectangle in a magnetic field and the voltage drop around a loop at a specific moment in time.
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