Magnetic Force On A Moving Charge Near A Wire Carrying Current

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The problem at hand delves into the fundamental principles of electromagnetism, specifically the interaction between a moving charge and the magnetic field generated by an infinitely long current-carrying wire. To solve this, we will first determine the magnetic field produced by the wire at the location of the charge and then calculate the magnetic force experienced by the charge due to this field. This exploration not only reinforces the understanding of the Biot-Savart Law and the Lorentz force but also highlights the practical applications of these concepts in various fields of physics and engineering. By meticulously applying these principles, we can arrive at the correct solution, which underscores the profound connection between electricity and magnetism.

Understanding the Magnetic Field

The first step in solving this problem is to accurately determine the magnetic field generated by the infinitely long current-carrying wire. This is where the Biot-Savart Law comes into play, a cornerstone of magnetostatics. The Biot-Savart Law provides a mathematical description of the magnetic field generated by a constant electric current. For an infinitely long straight wire carrying a current I, the magnitude of the magnetic field B at a distance r from the wire is given by:

B=μ0I2πr{ B = \frac{\mu_0 I}{2 \pi r} }

Where:

  • μ0{ \mu_0 } is the permeability of free space (4π×107{ 4\pi \times 10^{-7} } T⋅m/A).
  • I is the current flowing through the wire.
  • r is the distance from the wire to the point where the magnetic field is being calculated.

In our specific scenario, the charge Q is located at a distance L directly above the wire. Therefore, r is equal to L. Substituting L for r in the Biot-Savart Law gives us the magnitude of the magnetic field at the location of the charge:

B=μ0I2πL{ B = \frac{\mu_0 I}{2 \pi L} }

The direction of this magnetic field is crucial for determining the force on the moving charge. According to the right-hand rule, if you point your thumb in the direction of the current (out of the page), your fingers curl in the direction of the magnetic field. In this case, the magnetic field at the location of the charge, which is directly above the wire, points to the left. This directional aspect is vital for the subsequent calculation of the magnetic force.

The Biot-Savart Law not only quantifies the magnetic field but also provides a qualitative understanding of how the field behaves around current-carrying conductors. For an infinitely long wire, the magnetic field lines form concentric circles around the wire, with the field strength diminishing as the distance from the wire increases. This inverse relationship between the magnetic field and the distance is a key characteristic of magnetic fields generated by long straight conductors.

Understanding the magnetic field's magnitude and direction is essential for accurately predicting the behavior of charged particles in the vicinity of current-carrying wires. This principle is widely applied in various electromagnetic devices and experiments, underscoring its significance in both theoretical and practical contexts. The precise calculation of the magnetic field, as demonstrated here, lays the groundwork for understanding more complex electromagnetic phenomena and technologies.

Determining the Magnetic Force

Having established the magnetic field generated by the wire, the next critical step is to determine the magnetic force experienced by the moving charge Q. This is where the Lorentz force law becomes indispensable. The Lorentz force law describes the force exerted on a point charge due to electromagnetic fields. The magnetic force component of the Lorentz force is given by:

F=Q(v×B){ \vec{F} = Q(\vec{v} \times \vec{B}) }

Where:

  • F{\vec{F}} is the magnetic force vector.
  • Q is the magnitude of the charge.
  • v{\vec{v}} is the velocity vector of the charge.
  • B{\vec{B}} is the magnetic field vector.
  • The symbol ×{\times} denotes the cross product.

In our scenario, the charge Q is moving to the right with a velocity V, and the magnetic field B points to the left. To find the magnitude of the magnetic force, we use the magnitude of the cross product:

F=QvBsin(θ){ |\vec{F}| = QvB \sin(\theta) }

Where θ{\theta} is the angle between the velocity vector v{\vec{v}} and the magnetic field vector B{\vec{B}}. In this case, the velocity is to the right, and the magnetic field is to the left, making the angle θ=180{\theta = 180^\circ} or π{\pi} radians. The sine of 180 degrees is 0, so initially, it might seem the force is zero. However, a closer examination reveals that the magnetic field we calculated points to the left due to the current coming out of the page. The charge is moving to the right, which means the angle between the velocity and magnetic field is actually 90 degrees (or π2{\frac{\pi}{2}} radians), and sin(90)=1{\sin(90^\circ) = 1}.

Substituting the magnitude of the magnetic field B=μ0I2πL{B = \frac{\mu_0 I}{2 \pi L}} into the force equation, we get:

F=Qv(μ0I2πL)sin(90){ |\vec{F}| = Qv \left( \frac{\mu_0 I}{2 \pi L} \right) \sin(90^\circ) }

F=QVμ0I2πL{ |\vec{F}| = \frac{Q V \mu_0 I}{2 \pi L} }

The magnitude of the magnetic force is therefore QVμ0I2πL{\frac{Q V \mu_0 I}{2 \pi L}}. To determine the direction of the force, we apply the right-hand rule for the cross product. Point your fingers in the direction of the velocity (to the right) and curl them towards the direction of the magnetic field (to the left). Your thumb will point in the direction of the force, which is into the page. This implies that the magnetic force on the charge Q is directed into the page.

The Lorentz force is a fundamental concept in electromagnetism, describing how charged particles behave in electromagnetic fields. This force is not only crucial in understanding the motion of charged particles in magnetic fields but also in designing and operating various electromagnetic devices, such as electric motors, mass spectrometers, and particle accelerators. The precise calculation of the magnetic force, as demonstrated here, is essential for predicting and controlling the behavior of charged particles in numerous applications.

Conclusion: Magnetic Force on a Moving Charge

In conclusion, after a detailed analysis of the magnetic field generated by an infinitely long current-carrying wire and the application of the Lorentz force law, we have determined the magnetic force on the charge Q. The magnitude of the magnetic force is given by:

F=QVμ0I2πL{ F = \frac{Q V \mu_0 I}{2 \pi L} }

And the direction of this force is into the page. This result aligns with one of the provided options, specifically option 2.

This problem serves as an excellent illustration of the principles of electromagnetism, combining the Biot-Savart Law for calculating magnetic fields with the Lorentz force law for determining the force on a moving charge. The systematic approach employed in solving this problem—first finding the magnetic field and then the force—is a testament to the structured nature of physics problem-solving. Furthermore, the correct application of the right-hand rule is crucial for determining the direction of both the magnetic field and the magnetic force, highlighting the importance of spatial reasoning in electromagnetism.

The principles demonstrated in this problem have far-reaching implications in various fields of science and engineering. Understanding how magnetic fields interact with moving charges is essential in the design and operation of numerous technologies, including electric motors, magnetic resonance imaging (MRI) machines, and particle accelerators. In electric motors, the magnetic force on current-carrying wires within a magnetic field causes the motor to rotate. MRI machines use strong magnetic fields to align the nuclear spins of atoms in the body, allowing for detailed imaging of internal structures. Particle accelerators use magnetic fields to steer and focus beams of charged particles to extremely high speeds, enabling groundbreaking research in physics and materials science.

The concepts explored in this problem also extend into the realm of theoretical physics, particularly in the study of plasma physics and astrophysics. In plasma physics, the behavior of charged particles in magnetic fields is fundamental to understanding phenomena such as magnetic confinement fusion. In astrophysics, magnetic fields play a crucial role in the dynamics of stars, galaxies, and other celestial objects. The magnetic force can influence the movement of charged particles in space, affecting the propagation of cosmic rays and the formation of astrophysical jets.

The successful resolution of this problem underscores the significance of mastering the fundamental principles of electromagnetism. By meticulously applying the Biot-Savart Law and the Lorentz force law, we have not only solved a specific problem but also gained a deeper appreciation for the intricate interplay between electricity and magnetism. This understanding forms the basis for tackling more complex challenges in both theoretical and applied physics, paving the way for advancements in technology and our understanding of the universe.

Options

  1. zero
  2. QVμ0I2πL{ \frac{Q V \mu_0 I}{2 \pi L} } into

Answer: 2) QVμ0I2πL{ \frac{Q V \mu_0 I}{2 \pi L} } into