Homework #4
Conceptual Solution

1. (a) The direction of the magnetic field produced at the center of a current loop is given by the right hand rule: wrap your fingers around the loop, pointing in the direction of the current flow, and your thumb points in the direction of . Here for thumb and pointing inward, your fingers point clockwise around the loop.
2. (c) The direction of the magnetic moment of a current loop is given by the right hand rule, and is the same as the direction of the central magnetic field produced by the loop. Thus, since the loop's magnetic field points inward, its magnetic moment points inward also. Its magnitude is

   

3. (b) A magnetic field acts to align a bar magnet along the field lines, with it's north pole pointing in the direction of the field. Here the field is horizontal, pointing into the paper. Thus the magnet will align horizontally, with its north pole pointing into the paper.
4. (a) A straight wire in a magnetic field experiences the force , proportional to the vector cross product between and . If and are parallel, as in this case where both are horizontal vectors pointing into the paper, their vector cross product is zero. (Remember that the cross product involves the perpendicular component of one vector with respect to another, so vanishes when both vectors are parallel.) Thus a straight wire pointing in this direction experiences no force due to the magnetic field of the current loop.
5. (b) To cancel the inward magnetic field at the center of the current loop, the straight wire must produce an outward magnetic field of the same magnitude at that point. Of the three possibilities, only (b) can produce a magnetic field directed out of the page at the center of the loop. Each wire produces a magnetic field whose direction is tangential to a circle around the wire, going counterclockwise around that circle according to the right hand rule (thumb pointing along , fingers sweep along the circle in the local direction of ). Wire (a) (horizontal inward) produces pointing right directly above it, downward directly to its right, left directly below it, and up directly to its left. Thus to its right, in the middle of the current loop, it contributes a pointing downward, not outward. Wires (b) and (c), vertical upward, produce pointing left directly behind them, out directly to their left, right directly in front of them, and in directly to their right. Thus wire (b) produces an outward at the center of the loop, to its left, while wire (c) produces an inward at the center of the loop, to its right.

   

6. (a) The miniloop feels a torque of which has its maximal value, , when the magnetic moment lies perpendicular to . Here that maximal value is The magnetic moment aligns by pointing in the same direction as , which is here into the page. From the right hand rule relating the current in the miniloop to its magnetic moment, an inward corresponds to a miniloop in the plane of the paper with current flowing clockwise; that is, just like current in the large loop. A simpler way to say this is that the miniloop's points along the large loop's ; since each is related to its current loop by the right hand rule, the current loops must also be aligned and flowing in the same direction.

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