Question

In: Physics

A current of 16.0 mA is maintained in a single circular loop of 1.90 m circumference....

A current of 16.0 mA is maintained in a single circular loop of 1.90 m circumference. A magnetic field of 0.790 T is directed parallel to the plane of the loop.

(a) Calculate the magnetic moment of the loop.
mA · m2

(b) What is the magnitude of the torque exerted by the magnetic field on the loop?
mN · m

Solutions

Expert Solution

Concepts and reason

The main concepts required to solve this problem are the magnetic moment, current, magnetic field, area and torque.

Initially, write the equation for the magnetic moment of the loop, area of the loop and the toque exerted by the magnetic field in the loop. Use these equations and calculate the magnetic moment of the loop and the toque exerted by the magnetic field in the loop.

Fundamentals

The equation for the magnetic moment of the loop is,

μ=IA\mu = IA

Here, I is the current through the loop, and A is the area of the loop.

The equation for the area of the loop is,

A=πr2A = \pi {r^2}

Here, r is the radius of the loop.

The equation for the circumference of the circular loop is,

C=2πr{\rm{C}} = 2\pi r

Here, r is the radius of the loop.

The equation for the torque exerted by the magnetic field in the loop is,

τ=IAB\tau = IAB

Here, I is the current, A is the area of the loop and B is the magnetic field.

(a)

The equation for the circumference of the circular loop is,

C=2πr{\rm{C}} = 2\pi r

Rearrange the above equation for r.

r=C2πr = \frac{{\rm{C}}}{{2\pi }}

Substitute 1.90 m for C, and 3.14 for π\pi in above equation.

r=1.90m2(3.14)=0.3025m\begin{array}{c}\\r = \frac{{1.90{\rm{ m}}}}{{2\left( {3.14} \right)}}\\\\ = 0.3025{\rm{ m}}\\\end{array}

The equation for the area of the loop is,

A=πr2A = \pi {r^2}

Substitute 3.14 for π\pi , and 0.3025m0.3025{\rm{ m}} for r in above equation.

A=(3.14)(0.3025m)2=0.287m2\begin{array}{c}\\A = \left( {3.14} \right){\left( {0.3025{\rm{ m}}} \right)^2}\\\\ = 0.287{\rm{ }}{{\rm{m}}^2}\\\end{array}

The equation for the magnetic moment of the loop is,

μ=IA\mu = IA

Substitute 16.0mA16.0{\rm{ mA}} for I and 0.287m20.287{\rm{ }}{{\rm{m}}^2} for A in above equation.

μ=(16.0mA)(0.287m2)=4.58mAm2\begin{array}{c}\\\mu = \left( {16.0{\rm{ mA}}} \right)\left( {0.287{\rm{ }}{{\rm{m}}^2}} \right)\\\\ = 4.58\,\,{\rm{mA}} \cdot {{\rm{m}}^2}\\\end{array}

(b)

The equation for the torque exerted by the magnetic field in the loop is,

τ=IAB\tau = IAB

Substitute 16.0mA16.0{\rm{ mA}} for I, 0.287m20.287{\rm{ }}{{\rm{m}}^2} for A, and 0.790T0.790{\rm{ T}} for B in above equation.

τ=(16.0mA)(0.287m2)(0.790T)=3.62mNm\begin{array}{c}\\\tau = \left( {16.0{\rm{ mA}}} \right)\left( {0.287{\rm{ }}{{\rm{m}}^2}} \right)\left( {0.790{\rm{ T}}} \right)\\\\ = 3.62{\rm{ mN}} \cdot {\rm{m}}\\\end{array}

Ans: Part a

The magnetic moment of the loop is 4.58mAm24.58\,\,{\rm{mA}} \cdot {{\rm{m}}^2} .

Part b

The torque exerted by the magnetic field in the loop is 3.62mNm3.62{\rm{ mN}} \cdot {\rm{m}} .


Related Solutions

A current of 17.0 mA is maintained in a single circular loop with a circumference of...
A current of 17.0 mA is maintained in a single circular loop with a circumference of 2.00 m. A magnetic field of 0.800 T is directed parallel to the plane of the loop. (a) Calculate the magnetic moment of the loop. (b) What is the magnitude of the torque exerted on the loop by the magnetic field?
Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 168 cmcm...
Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 168 cmcm , but its circumference is decreasing at a constant rate of 10.0 cm/s due to a tangential pull on the wire. The loop is in a constant uniform magnetic field of magnitude 0.800 T , which is oriented perpendicular to the plane of the loop. Assume that you are facing the loop and that the magnetic field points into the loop. Find the magnitude...
Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 167 cm...
Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 167 cm , but its circumference is decreasing at a constant rate of 14.0 cm/s due to a tangential pull on the wire. The loop is in a constant uniform magnetic field of magnitude 0.900 T , which is oriented perpendicular to the plane of the loop. Assume that you are facing the loop and that the magnetic field points into the loop. Find the magnitude...
A square current loop 5.5 cm on each side carries a 470 mA current. The loop...
A square current loop 5.5 cm on each side carries a 470 mA current. The loop is in a 0.70 T uniform magnetic field. The axis of the loop, perpendicular to the plane of the loop, is 30 degrees away from the field direction. What is the magnitude of the torque on the current loop? 2 sig fig.
A circular loop of radius 1.79 m is mounted on top of a truck. Initially the...
A circular loop of radius 1.79 m is mounted on top of a truck. Initially the truck is at rest oriented so that a vector normal to the loop points north. At this location, the Earth’s magnetic field has a magnitude of 49.6 μT and a direction of due north, 63° below the horizontal. Over the course of 9.0 seconds, the truck accelerates north, so that at the end of the time interval, the truck is moving at 14.0 m/s...
A conducting single-turn circular loop with a total resistance of 7.50 Ω is placed in a...
A conducting single-turn circular loop with a total resistance of 7.50 Ω is placed in a time-varying magnetic field that produces a magnetic flux through the loop given by ΦB = a + bt2 − ct3, where a = 8.00 Wb, b = 15.5 Wb/s−2, and c = 7.50 Wb/s−3. ΦB is in webers, and t is in seconds. What is the maximum current induced in the loop during the time interval t = 0 to t = 1.55 s?...
Intro to Electricity and Magnetism Derive the magnetic vector potential of a circular loop of current....
Intro to Electricity and Magnetism Derive the magnetic vector potential of a circular loop of current. Then calculate the magnetic field from your result for the magnetic vector potential.
A circular loop of wire having a radius of 6.0 cm carries a current of 0.14...
A circular loop of wire having a radius of 6.0 cm carries a current of 0.14 A. A unit vector parallel to the dipole moment of the loop is given by 0.40i -0.92j. If the loop is located in a magnetic field given by B = (0.40 T)i + (0.35 T)k, A) find the magnitude of the magnetic dipole moment of the loop. B) Find the i component of the torque on the loop. C) Find the j component of...
Two circular current-carrying loops of wire are shown in the drawing. The inner loop has a...
Two circular current-carrying loops of wire are shown in the drawing. The inner loop has a radius of R0 and carries a current I0, while the outer loop has a radius of 2R0 and carries a current of 4I0. The currents are in opposite directions. If R0 = 0.20 m and I0 = 1.9 A, determine the magnitude and direction of the net magnetic field at the center of the two loops.
A circular wire loop of radius r = 0.25 m and resistance R = 11.1 Ω...
A circular wire loop of radius r = 0.25 m and resistance R = 11.1 Ω rotates about a shaft through its diameter at a constant rate of f = 5.8 Hz in a uniform B = 0.49-T magnetic field directed perpendicular to the rotation axis. The plane of the loop is perpendicular to the magnetic field at time t = 0. a) Find the expression for the time-dependent magnetic flux through the loop. Φ = b) Find the value...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT