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Home » Express \( w \) in terms of \( l \): - Portal da AcÃºstica

Express \( w \) in terms of \( l \): - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question:** An electrical engineer analyzes the magnetic flux through a coil whose area vector \( \vec{A} \) rotates in the \( xy \)-plane. If \( \vec{A}(t) = A_0 \cos(\omega t) \hat{\mathbf{x}} + A_0 \sin(\omega t) \hat{\mathbf{y}} \) and the magnetic field is \( \vec{B} = B_0 \hat{\mathbf{z}} \), compute the time average of the magnetic flux \( \Phi_B = \vec{B} \cdot \vec{A}(t) \) over one period.

📌 The magnetic flux is:

📌 \Phi_B(t) = \vec{B} \cdot \vec{A}(t) = B_0 \hat{\mathbf{z}} \cdot \left( A_0 \cos(\omega t) \hat{\mathbf{x}} + A_0 \sin(\omega t) \hat{\mathbf{y}} \right) = 0

📌 since \( \hat{\mathbf{z}} \) is perpendicular to both \( \hat{\mathbf{x}} \) and \( \hat{\mathbf{y}} \), and the dot product with zero components is zero.

📌 Thus, \( \Phi_B(t) = 0 \) for all \( t \). The time average over one period \( T = \frac{2\pi}{\omega} \) is:

📌 \langle \Phi_B \rangle = \frac{1}{T} \int_0^T \Phi_B(t) \, dt = \frac{1}{T} \int_0^T 0 \, dt = 0

📌 The time-averaged magnetic flux is:

📌 Question:** A cultural anthropologist observes a ceremonial circle divided into 12 equal segments, each representing a lunar phase. If a participant starts at segment 1 and jumps \( n \) segments clockwise each day, where \( n \equiv 7 \pmod{12} \), after how many days will the participant return to segment 1?

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