Question

A 50 -turn rectangular coil of wire with dimensions \(10.0 \mathrm{~cm}\) by \(20.0 \mathrm{~cm}\) lies in a horizontal plane, as shown in the figure. The axis of rotation of the coil is aligned north and south. It carries a current \(i=1.00 \mathrm{~A}\) and is in a magnetic field pointing from west to east. A mass of \(50.0 \mathrm{~g}\) hangs from one side of the coil. Determine the strength the magnetic field has to have to keep the coil in the horizontal orientation.

Short answer

Answer: The strength of the magnetic field has to be approximately \(0.123 \mathrm{T}\) to keep the coil in the horizontal orientation.

Step-by-step solution

Calculate the torque due to the magnetic field on the coil

To calculate the torque due to the magnetic field, we can use the formula: \(\tau_B = nABi\sin{\theta}\) where \(\tau_B\) is the torque due to the magnetic field, n is the number of turns in the coil, A is the area of the coil, B is the magnetic field strength, i is the current in the coil, and \(\theta\) is the angle between the magnetic field and the normal to the plane of the coil. In this case, the angle \(\theta\) between the magnetic field and the normal to the plane of the coil is \(90^\circ\), so \(\sin{\theta} = 1\). So the torque due to the magnetic field can be simplified to: \(\tau_B = nABi\).

Calculate the torque due to gravity on the coil

To calculate the torque due to gravity acting on the coil, we can use the formula: \(\tau_g = mgd\) where \(\tau_g\) is the torque due to gravity, m is the mass hanging from the coil, g is the acceleration due to gravity, and d is the distance from the axis of rotation to the center of mass of the hanging mass. In this case, the distance from the axis of rotation to the center of mass of the hanging mass is equal to the width of the coil, which is \(10.0 \mathrm{~cm} = 0.1 \mathrm{~m}\).

Set the torques equal to each other and solve for B

To keep the coil in a horizontal orientation, the torque due to the magnetic field must be equal to the torque due to gravity: \(\tau_B = \tau_g\) So we have: \(nABi = mgd\) Now we can solve for B: \(B = \frac{mgd}{nAi}\)

Plug in the given values and calculate B

Now we have all the necessary values: n = 50 A = \((10.0 \mathrm{~cm} \times 20.0 \mathrm{~cm}) = 0.02 \mathrm{~m^2}\) i = 1.00 A m = \(50.0 \mathrm{~g} = 0.05 \mathrm{~kg}\) g = \(9.81 \mathrm{~m/s^2}\) d = \(10.0 \mathrm{~cm} = 0.1 \mathrm{~m}\) Plugging these values into our formula for B: \(B = \frac{(0.05 \mathrm{~kg})(9.81 \mathrm{~m/s^2})(0.1 \mathrm{~m})}{(50)(0.02 \mathrm{~m^2})(1.00 \mathrm{~A})}\) Finally, calculating B: \(B \approx 0.123 \mathrm{T}\) The strength of the magnetic field has to be approximately \(0.123 \mathrm{T}\) to keep the coil in the horizontal orientation.

Key concepts

Torque Due to Magnetic Field

Understanding the torque due to a magnetic field is essential when studying magnetic effects on current-carrying conductors. Torque can be thought of as a rotational force that causes an object to twist around an axis. In the realm of electromagnetism, when a current-carrying loop is placed in a magnetic field, it experiences a torque because of the Lorentz forces acting on the charges moving through the loop. The magnitude of this torque is given by the formula \(\tau_B = nABi\sin{\theta}\), where each symbol represents a contributing factor to the torque.

The number of turns in the coil (n) and the area of the coil (A) are directly proportional to the torque, meaning more turns or a larger area would result in a greater twisting force. The current (i) flowing through the coil also directly affects the torque, with a higher current producing a stronger interaction with the magnetic field. The angle \(\theta\) between the magnetic field \(\mathbf{B}\) direction and the normal (perpendicular) to the coil uniquely influences the torque through the sine function; torque is maximized when this angle is 90 degrees.

Applying this knowledge, students can better visualize how the physical setup and orientation of the coil within the magnetic field impact the resulting torque.

Torque Due to Gravity

Torque isn't only the purview of magnetic fields; gravity also exerts a torque on objects, which can be calculated to understand balancing forces in physics problems. The torque due to gravity \(\tau_g\) arises when a force (in this case, the force of gravity) is applied at a distance (d) from an object's pivot point or axis of rotation. As described by the equation \(\tau_g = mgd\), the mass (m) of the object and the acceleration due to gravity (g) play crucial roles.

The distance (d) from the pivot point to the center of mass where gravity effectively acts, is of particular importance. When this distance is increased, the gravitational torque increases, creating a stronger rotational effect. For example, in our problem, a mass hanging off one side of the coil means that gravity is exerting a torque that tries to rotate the coil downward. This concept allows students to analyze situations where balance under gravity's influence is a factor, such as when determining the conditions for the stability of structures or equipment.

Equilibrium in Magnetic Fields

Equilibrium within magnetic fields pertains to the state where all forces and torques are balanced, resulting in no net movement or rotation of the system. For an object to remain at rest or to retain a constant velocity, the torques acting upon it (due to both magnetic fields and gravity in the case of our example problem) must cancel each other out. This concept is expressed using the equilibrium equation \(\tau_B = \tau_g\).

Achieving equilibrium is a delicate balance, hinging on the exact values of magnetic field strength, coil dimensions, current, and the mass and positioning of any weights attached. In the educational exercise provided, the key to maintaining a horizontal coil involves precisely calibrating the strength of the magnetic field such that its torque perfectly counters the torque from the hanging mass. By manipulating the variables accurately, students can master the concept of equilibrium, a principle that's vital to systems ranging from simple classroom demonstrations to advanced engineering designs.

Physics Problem Solving

Physics problem solving is a multifaceted skill that involves understanding concepts, applying mathematical relationships, and executing logical steps to arrive at a solution. As seen in the step-by-step explanation of the given exercise, breaking the problem into digestible parts is a key strategy. Initially, we determine the torques due to magnetic fields and gravity separately, using their respective formulas. Then, we set these torques equal to establish equilibrium and solve for the unknown, which is the strength of the magnetic field in this case.

Success in problem-solving requires not only mathematical proficiency but also a deep comprehension of how different physical principles interact with each other. This approach aids students in building a holistic understanding of physics, from the micro-scale interactions of particles and fields to the macro-scale laws governing objects in motion. Encouraging students to visualize each step involved, especially transitions between different forms of energy or forces, enhances their ability to tackle complex physics problems with confidence and ingenuity.