7ssm The maximum torque experienced by a coil in a 0.75 -T magnetic field is 8.4 ×10^-4 N . The

7ssm The maximum torque experienced by a coil in a 0.75 -T...

Key Concepts

Magnetic Torque on a Current Loop

When a current loop is placed in a magnetic field, it experiences a torque due to the interaction between the magnetic field and the current flowing in the loop. The torque tends to align the plane of the loop perpendicular to the magnetic field, and its maximum value is achieved when the normal to the loop is perpendicular to the field. This phenomenon is central to devices like electric motors and galvanometers.

Magnetic Moment of a Coil

The magnetic moment of a coil is a measure of its tendency to align with a magnetic field and is defined as the product of the current and the area of the coil. For a single-turn coil, the magnetic moment is given by I multiplied by A, where I is the current and A is the area. This concept links the electrical properties of the coil to its geometric dimensions.

Relationship Between Coil Geometry and Magnetic Moment

For a circular coil, the area is related to its radius (or its circumference) by the formula A = ?r². When the coil’s length (or the circumference in the case of a single-turn coil) is known, this relationship can be manipulated to translate between the geometric properties and the magnetic moment. This is crucial when one must determine one geometric parameter (like the length of the wire) from magnetic and electrical measurements.

Transcript

00:01 In this problem, we're given information about the torque acting on a circular coil consisting of one turn due to an external magnetic field while current is running through that coil.

00:14 So we're given something about the maximum torque that is acting on our circle.

00:20 And so we know that the torque acting on a coil of wire due to a current running through that wire, current running through that coil being in an external magnetic field, it's given by the number of turns in that coil, multiplied by the current traveling through that coil, multiplied by the cross -sectional area of the coil, multiplied by the magnetic field that the coil is experiencing, multiplied by sign of the angle between the normal to the surface area of a coil and the external magnetic field.

00:51 But since we're asked about or given information about the maximum torque, that is when our sine phi part of our equation goes to 1 because the maximum torque occurs when the normal and the magnetic fields are perpendicular, so that phi is 90.

01:07 So we're really only dealing with the equation of n, i, a, b.

01:12 Now we're asked an interesting thing.

01:14 We're asked to find out what is the length of the wire that this coil is made.

01:19 So basically we just have a singular one loop coil.

01:24 That's it.

01:24 So if one loop, we have n equals 1, right? and we know that the area, the cross -sectional area of our circular loop, which is given by pi r squared, right? say we have this circle as radius r can be turned into an expression in terms of not the area of our circle, but the circumference of our circle.

01:46 So remember that the circumference of our circles, the length all the way around is given by 2 pi r, and therefore r is equal to the circumference of our circle.

01:54 Of our circle divided by 2 pi.

01:57 Plug that into here, right? we get that the area as an expression, not in terms of the radius of our circle, but the circumference of our circle is given by the circumference squared divided by 4 pi.

02:14 Okay, cool.

02:15 So now, right, we want to find that circumference.

02:17 You want to find how long this circle is...

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