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Magazine Chapter 5: Problem 5
Calculate the magnetic field strength needed on a 200-turn square loop \(20.0 \mathrm{~cm}\) on a side to create a maximum torque of \(300 \mathrm{~N} \cdot \mathrm{m}\) if the loop is carrying \(25.0 \mathrm{~A}\).
Short Answer
Expert verified
Magnetic field strength needed is 1.5 T.
Step by step solution
01
Identify the known values
We have a 200-turn square loop with sides measuring \(20.0\,\mathrm{cm}\), carrying a current of \(25.0\,\mathrm{A}\), and needing to generate a maximum torque of \(300\,\mathrm{N}\cdot\mathrm{m}\).
02
Convert the side length to meters
Convert the side length from centimeters to meters: \(20.0\,\mathrm{cm} = 0.20\,\mathrm{m}\).
03
Calculate the area of the loop
The area \(A\) of a square is given by the formula \(A = \text{side}^2\). Substituting the side length gives \(A = (0.20)^2 = 0.04\,\mathrm{m}^2\).
04
Use the torque formula
The torque \(\tau\) on a coil in a magnetic field is given by \(\tau = nIAB \sin \theta\), where \(n\) is the number of turns, \(I\) is the current, \(A\) is the area, \(B\) is the magnetic field strength, and \(\theta\) is the angle between the normal to the coil and the magnetic field. For maximum torque, \(\sin \theta = 1\), therefore, \(\tau = nIAB\).
05
Solve for the magnetic field strength \(B\)
Rearrange the formula to solve for \(B\): \[ B = \frac{\tau}{nIA} \] Substituting the values: \[ B = \frac{300}{200 \times 25 \times 0.04} = \frac{300}{200} = 1.5\,\mathrm{T} \]
06
Answer verification step
Recalculate to ensure everything checks out:\[ B = \frac{300}{200 \times 25 \times 0.04} = \frac{300}{200} = 1.5\,\mathrm{T} \] The calculation confirms that the magnetic field strength is \(1.5\,\mathrm{T}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Torque Calculation
Torque is a measure of how much a force acting on an object causes that object to rotate. It's fundamental in understanding rotational motion. In our scenario, torque is generated in a coil placed in a magnetic field. The formula for torque (\(\tau\)) in this context is:
- \(\tau = n\times I \times A \times B \times \sin \theta\)
- \(n\) represents the number of turns in the coil,
- \(I\) is the current flowing through the coil,
- \(A\) is the area of the coil,
- \(B\) is the magnetic field strength, and
- \(\theta\) is the angle between the magnetic field and the normal to the plane of the coil.
Square Loop
A square loop is simply a loop of wire where each side has equal length. In electromagnetism, square loops are frequently used in theoretical problems because they have straightforward geometric properties. Each side of our square loop is \(20.0\,\mathrm{cm}\), which is \(0.20\,\mathrm{m}\) when converted to meters.
The area \(A\) of the square loop is calculated as the side length squared, \(A = \text{side}^2\). For a square loop with a side of \(0.20\,\mathrm{m}\), the area will be \(0.20 \times 0.20 = 0.04\,\mathrm{m}^2\). This area quantitatively attributes to how the magnetic field interacts with the loop to produce torque.
The area \(A\) of the square loop is calculated as the side length squared, \(A = \text{side}^2\). For a square loop with a side of \(0.20\,\mathrm{m}\), the area will be \(0.20 \times 0.20 = 0.04\,\mathrm{m}^2\). This area quantitatively attributes to how the magnetic field interacts with the loop to produce torque.
Coil in Magnetic Field
When a coil or a loop of wire is placed in a magnetic field, it experiences a force that can cause it to rotate if the coil is carrying an electrical current. This is because magnetic fields exert a force on moving charges according to the right-hand rule.
The interaction between the magnetic field and the current generates torque, causing rotational motion. This principle is widely applied in electrical motors and generators where coils in magnetic fields convert electrical energy into mechanical motion, or vice versa. The critical point is the perpendicular interaction default for maximal torque, enhancing the efficiency of energy transfer.
The interaction between the magnetic field and the current generates torque, causing rotational motion. This principle is widely applied in electrical motors and generators where coils in magnetic fields convert electrical energy into mechanical motion, or vice versa. The critical point is the perpendicular interaction default for maximal torque, enhancing the efficiency of energy transfer.
Current in a Coil
Current refers to the flow of electric charge through a conductor, such as a coil. In our problem, the coil carries a current of \(25.0\,\mathrm{A}\). The presence of current in the wire creates a magnetic moment, which interacts with the external magnetic field.
The strength of the current, combined with the number of turns in the coil, significantly influences the torque produced.
The strength of the current, combined with the number of turns in the coil, significantly influences the torque produced.
- The greater the current, or the more coils involved, the more considerable the torque generated, increasing the effectiveness of the coil as an actuator or motor component.
- This principle highlights the importance of current magnitude in applications requiring significant force or motion.
