Explain it solution ---**Question Stem:** A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass m and radius r and it is in a uniform vertical magnetic field B₀, as shown in the figure. Initially, it hangs vertically downwards, because of acceleration due to gravity g, on two conducting supports at P and Q. When a current I is passed through the loop, the loop turns about the line PQ by an angle θ given by **Options:** (A) tan θ = πrIB₀/(mg) (B) tan θ = 2πrIB₀/(mg) (C) tan θ = πrIB₀/(2mg) (D) tan θ = mg/(πrIB₀) **Chart/Diagram Description:** * **Type:** Schematic diagram illustrating a physics setup. * **Main Elements:** * A horizontal line segment connects points P and Q. * From points P and Q, vertical lines extend downwards, attached to support structures (indicated by hatched lines). * A circular loop is suspended below the line PQ. The loop's diameter aligns vertically, centered between P and Q. The line PQ appears to be a tangential line to the top of the loop where its two ends are attached. * A label 'r' with a dashed line indicates the radius of the circular loop, extending from the center vertically upwards to the top point where the loop meets the line PQ. * A label 'm' is placed near the circular loop, indicating its mass. * An arrow pointing vertically upwards is labeled 'B₀', indicating a uniform vertical magnetic field. * An arrow pointing vertically downwards is labeled 'g', indicating the acceleration due to gravity. * The diagram represents the initial state before the current is passed. The line PQ is the axis of rotation after the current is passed, resulting in a tilt by angle θ. The angle θ itself is not explicitly shown in this initial diagram but is mentioned in the question stem as the rotation angle about PQ.

视频信息

答案文本 复制

视频字幕 复制

We have a circular wire loop with mass m and radius r suspended from two supports P and Q. The loop is in a uniform vertical magnetic field B zero. When current I flows through the loop, it experiences a magnetic force that causes it to rotate by angle theta about the axis P Q. We need to find the expression for tangent theta. Let's analyze the forces acting on the current loop. First, there's the gravitational force mg acting downward at the center of mass. Second, when current flows through the loop in the magnetic field, each segment experiences a magnetic force. The total magnetic force on the loop is I times B zero times the total length, which is I B zero times two pi r. This magnetic force acts horizontally and creates a torque about the rotation axis P Q. Now let's analyze the torques about the rotation axis P Q. The gravitational force creates a restoring torque equal to mg times r times sine theta. The magnetic force creates a torque equal to the magnetic force times the lever arm r. At equilibrium, these torques must balance. This gives us mg times r times sine theta equals I B zero times two pi r times r. We have a circular wire loop with mass m and radius r suspended from a horizontal axis P Q. The loop is in a uniform vertical magnetic field B zero. When current I flows through the loop, it experiences a magnetic force that causes it to rotate about the P Q axis by an angle theta. Let's analyze the forces. The gravitational force mg acts downward at the center of mass. The magnetic force on each current element is given by I d l cross B. For a circular loop, the total magnetic force is I times two pi r times B zero, acting horizontally. The gravitational torque about P Q is mg times r times sine theta. The magnetic torque is the magnetic force times the lever arm r. At equilibrium, these torques are equal. At equilibrium, the loop tilts by angle theta. The gravitational force creates a restoring torque proportional to r sine theta, while the magnetic force creates a torque proportional to the current and magnetic field. The equilibrium angle is determined when these torques balance. Let's solve the equilibrium equation step by step. Starting from mg times r times sine theta equals I B zero times two pi r times r, we can cancel one r from both sides to get mg sine theta equals I B zero times two pi r. Solving for sine theta, we get sine theta equals I B zero times two pi r divided by mg. For small angles, sine theta is approximately equal to tangent theta. Therefore, tangent theta equals two pi r I B zero divided by mg. This matches option B. To summarize: A current-carrying loop in a magnetic field experiences a torque that rotates it until equilibrium. The balance between magnetic and gravitational torques gives us tangent theta equals two pi r I B zero divided by mg, which is option B. To summarize: A current-carrying loop in a magnetic field experiences a torque that rotates it until equilibrium. The balance between magnetic and gravitational torques gives us tangent theta equals two pi r I B zero divided by mg, which is option B.