Let's answer each part step by step, explaining reasoning at each stage: --- ## a. Direction of Current for Clockwise Rotation **Step 1:** The torque (\(\vec{\tau}\)) on a current loop in a magnetic field is given by: \[ \vec{\tau} = \vec{\mu} \times \vec{B} \] where \(\vec{\mu}\) is the magnetic moment (\(\vec{\mu} = I \cdot \vec{A}\)), and \(\vec{B}\) is the magnetic field (up the page). **Step 2:** To make the loop spin **clockwise** as seen from your view, the torque must act to decrease \(\theta\). - Using the right-hand rule: For the **top left** of the loop, to produce clockwise torque, the current should be **into the page**. - For the **bottom right**, the current should be **out of the page**. **Explanation:** This current direction creates a magnetic moment that, when crossed with the upward field, produces a clockwise torque. --- ## b. Magnitude of the Loop's Torque **Step 1:** The magnitude of torque is: \[ \tau = \mu B \sin \theta = (I A) B \sin \theta \] **Step 2:** At the initial angle \(\theta_\), the torque is: \[ \tau = IAB \sin \theta_ \] **Explanation:** The torque is maximum when \(\theta = 90^\circ\) and zero when \(\theta = \) or \(180^\circ\). --- ## c. At What \(\theta\) Do We Flip the Current (First Time)? **Step 1:** The torque will switch direction when the angle \(\theta\) crosses \(^\circ\) (vertical position). **Step 2:** **At \(\theta = ^\circ\) (loop is vertical),** flip the current to keep the torque in the same (clockwise) direction. **Explanation:** If the current isn't flipped, the torque would reverse and rotate the loop the other way (counter-clockwise). --- ## d. At What \(\theta\) Do We Flip Again? **Step 1:** After flipping at \(\theta = ^\circ\), the loop will rotate to \(\theta = -180^\circ\) (other vertical position). **Step 2:** **At \(\theta = -180^\circ\),** flip the current again to keep torque clockwise. **Explanation:** Each time the loop passes through the vertical position, the current must be flipped to maintain the same direction of rotation. --- ## e. If We Never Flip the Current, What Is the Maximum \(\theta\)? **Step 1:** If the current is never flipped, the torque will try to restore the loop to the nearest equilibrium position. **Step 2:** The loop will reach \(\theta = ^\circ\) (vertical), then the torque will reverse, making the loop start rotating counter-clockwise. **Final Answer:** The maximum value of \(\theta\) before reversing is **\(\theta = ^\circ\)**. --- # **Summary of Answers** a. **Current direction:** Into the page at top left, out of the page at bottom right (using right-hand rule for clockwise torque). b. **Torque magnitude:** \(\tau = IAB \sin \theta_\) c. **First flip:** At \(\theta = ^\circ\) (loop vertical) d. **Second flip:** At \(\theta = -180^\circ\) (loop vertical, other side) e. **Maximum \(\theta\) if never flipped:** \(\theta = ^\circ\)

## 1. Analysis of the Current Loop in Magnetic Field ### a. Direction of Current for Clockwise Rotation **Step 1:** To determine the direction of current that will make the loop rotate clockwise, we apply the right-hand rule. The torque (\(\vec{\tau}\)) on a current loop is given by: \[ \vec{\tau} = \vec{\mu} \times \vec{B} \] where \(\vec{\mu} = I \cdot \vec{A}\) is the magnetic moment and \(\vec{B}\) is the magnetic field. **Step 2:** For the loop to rotate clockwise, the torque must act to decrease the angle \(\theta\). Therefore: - At the **top left** of the loop, the current should flow **into the page**. - At the **bottom right**, the current should flow **out of the page**. **Explanation:** This configuration produces a magnetic moment that, when crossed with the upward magnetic field, generates a clockwise torque. --- ### b. Magnitude of the Loop's Torque **Step 1:** The magnitude of the torque acting on the loop can be calculated using the formula: \[ \tau = \mu B \sin \theta = (I A) B \sin \theta \] **Step 2:** At the initial angle \(\theta\), the torque is given by: \[ \tau = IAB \sin \theta \] **Explanation:** The torque reaches its maximum value when \(\theta = 90^\circ\) and becomes zero when \(\theta = 0^\circ\) or \(\theta = 180^\circ\). --- ### c. Angle for First Current Flip **Step 1:** The torque begins to reverse when the angle \(\theta\) approaches \(0^\circ\) (the vertical position). **Step 2:** **At \(\theta = 0^\circ\)** (loop is vertical), the current must be flipped to maintain clockwise torque. **Explanation:** If we do not flip the current at this position, the resulting torque will change direction, causing the loop to start rotating counter-clockwise. --- ### d. Angle for Second Current Flip **Step 1:** After the first flip at \(\theta = 0^\circ\), the loop will continue to rotate and will reach \(\theta = 180^\circ\) (the vertical position on the other side). **Step 2:** **At \(\theta = 180^\circ\)**, we must flip the current again to keep the torque in the clockwise direction. **Explanation:** Similar to the first flip, failing to flip the current here will cause the torque to reverse, leading to counter-clockwise rotation. --- ### e. Maximum Angle Before Reversal Without Flipping Current **Step 1:** If the current is never flipped, the loop will reach a maximum angle where the torque balances and reverses. **Step 2:** The maximum angle \(\theta\) before the loop starts rotating counter-clockwise is **\(\theta = 90^\circ\)**. **Final Answer:** The maximum angle of rotation before the loop turns back is **\(\theta = 90^\circ\)**. --- # **Summary of Answers** - a. **Current direction:** Into the page at the top left, out of the page at the bottom right. - b. **Torque magnitude:** \(\tau = IAB \sin \theta\). - c. **First flip:** At \(\theta = 0^\circ\) (loop vertical). - d. **Second flip:** At \(\theta = 180^\circ\) (loop vertical, other side). - e. **Maximum \(\theta\) if never flipped:** \(\theta = 90^\circ\).