Question

A straight wire with a constant current running through it is in Earth's magnetic field, at a location where the magnitude is \(0.43 \mathrm{G}\). What is the minimum current that must flow through the wire for a 10.0 -cm length of it to experience a force of \(1.0 \mathrm{~N} ?\)

Short answer

Answer: The minimum current required is 2.33 x 10^4 A.

Step-by-step solution

Write down the given information and formula

The given information is: - Length of the wire section (L): \(10.0 \, \mathrm{cm} = 0.1 \, \mathrm{m}\) - Earth's magnetic field (B): \(0.43\, \mathrm{G} = 0.43 \times 10^{-4} \, \mathrm{T}\) - Force (F): \(1.0\, \mathrm{N}\) The formula for the magnetic force on the wire is: \(F = I * L * B * \sin(\theta)\)

Determine the angle between the current and magnetic field

Since we need the minimum current, the angle between the current and the magnetic field should be such that the magnetic force is maximized. The function \(\sin(\theta)\) is maximized when \(\theta = 90 ^\circ\). So, we have: \(\theta = 90 ^\circ\)

Substitute values into the formula and solve for current

Now we will substitute the given values and the angle into the magnetic force formula to find the minimum current (I): \(1.0\, \mathrm{N} = I * 0.1\, \mathrm{m} * (0.43 \times 10^{-4}\, \mathrm{T}) * \sin(90 ^\circ)\) Since \(\sin(90^\circ) = 1\), the equation simplifies to: \(1.0\, \mathrm{N} = I * 0.1\, \mathrm{m} * (0.43 \times 10^{-4}\, \mathrm{T})\) Now, we can solve for the current (I): \(I = \frac{1.0\, \mathrm{N}}{0.1\, \mathrm{m} * (0.43 \times 10^{-4}\, \mathrm{T})} = \frac{1.0}{0.1 * 0.43 \times 10^{-4}}\, \mathrm{A}\) \(I = 2.33 \times 10^4\, \mathrm{A}\) Therefore, the minimum current required for the 10 cm section of the wire to experience a force of \(1.0\, \mathrm{N}\) is \(2.33 \times 10^4\, \mathrm{A}\).

Key concepts

Lorentz Force

The Lorentz force is a fundamental principle explaining how a magnetic field interacts with electric charges in motion. It refers to the force experienced by a charged particle or, by extension, a current-carrying wire, when it's in the presence of magnetic and electric fields. The formula for the Lorentz force on a wire is expressed as (F = I * L * B * \(sin(\theta)\)), where F is the magnetic force, I is the current, L is the length of the wire, B is the magnetic field strength, and \(\theta\) is the angle between the direction of the current and the magnetic field.

In the context of the exercise provided, the Lorentz force calculates the interaction between Earth's magnetic field and the electrical current running through a wire. Understanding this concept allows us to determine the energy and direction of the force acting on the wire, and by manipulating variables like the current, we can alter the magnitude of this force.

Earth's Magnetic Field

Earth's magnetic field, emanating largely from its core, is a protective and dynamic force that extends far out into space. While this field is most commonly recognized for its role in guiding compasses, it also has critical implications for many scientific and technological applications, including the scenario described in our exercise.

The strength of Earth's magnetic field at the surface typically ranges between about 0.25 to 0.65 Gauss, with variations depending on location and altitude. In the exercise, the wire is situated in a part of Earth's magnetic field that has a strength of 0.43 Gauss. It's important to note that the magnetic field is usually described in Tesla (T) in the SI unit system, and since 1 Gauss is equivalent to \(10^{-4}\) Tesla, students need to convert this measure appropriately to apply the Lorentz force equation.

Current and Magnetic Field Relationship

Understanding the relationship between current and magnetic field is vital in electromagnetism. The direction of the magnetic force on a current-carrying wire is perpendicular both to the direction of the magnetic field and to the current itself, as stated by the right-hand rule. This three-dimensional relationship is encapsulated in the Lorentz force equation mentioned earlier.

The magnitude of the force is directly proportional to the current, length of the wire, magnetic field strength, and the sine of the angle between the wire and the magnetic field (\(sin(\theta)\)). For the maximum force to be experienced, this angle must be 90 degrees, making the sine function equal to 1. This relationship is crucial when solving for the unknown current in the exercise, as we assume the ideal condition where this angle is 90 degrees to calculate the minimum current required to achieve a certain force.

Magnetic Field Strength

Magnetic field strength indicates the intensity of a magnetic field at a given point and is a key factor in determining the force exerted on a moving charged particle or current-carrying conductor. It's denoted by the symbol B and measured in Teslas (T) in the International System of Units (SI). Higher magnetic field strength results in a greater force when all other factors in the Lorentz force equation remain constant.

In the provided exercise, the wire's interaction with Earth's magnetic field is quantified to demonstrate the direct correlation between the magnetic field strength and the force experienced by the wire. If all other conditions remain the same, a stronger magnetic field would mean a greater force on the wire for any given amount of current.