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Chapter 7: Problem 38

Assuming that "Coulomb"s law" for magnetic charges \(\left(q_{m}\right)\) reads $$ \mathbf{F}=\frac{\mu_{0}}{4 \pi} \frac{q_{m_{1}} q_{m_{2}}}{{2}^{2}} \varepsilon_{1} $$ work out the force law for a monopole \(q_{n}\) moving with velocity \(v\) through electric and magnetic fields \(\mathbf{E}\) and \(\mathbf{B}\). \({ }^{26}\)

Short Answer

Expert verified
The force on a magnetic monopole in electric and magnetic fields is: \( \mathbf{F} = q_m \mathbf{B} - q_m\mathbf{v} \times \mathbf{E} \).

Step by step solution

01

Understanding the Problem

We need to derive the force experienced by a magnetic monopole moving through electric and magnetic fields. We will employ concepts from electromagnetism, specifically relating to electric charges and adapt them for magnetic monopoles.
02

Identify the Analogous Electric Charge Force

The force on an electric charge due to electric and magnetic fields is given by the Lorentz force law: \( \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \), where \( q \) is the electric charge, \( \mathbf{E} \) is the electric field, and \( \mathbf{B} \) is the magnetic field.
03

Define the Force Law for a Magnetic Monopole

In analogy to the Lorentz force law for electric charges, the force on a magnetic monopole \( q_m \) can be expressed as: \( \mathbf{F} = q_m(\mathbf{B} - \mathbf{v} \times \mathbf{E}) \). This is because the roles of \( \mathbf{E} \) and \( \mathbf{B} \) interchanges for the monopole, and 'electric charge' \( q \) becomes the 'magnetic charge' \( q_m \).
04

Incorporate Coulomb's Law for Magnetic Charges

Given Coulomb's law for magnetic charges, \( \mathbf{F} = \frac{\mu_{0}}{4 \pi} \frac{q_{m_{1}} q_{m_{2}}}{r^2} \mathbf{\varepsilon_{1}} \), for the scenario where \( q_{m_2} \) is within the field created by \( q_{m_1} \) and obeying similar interactions.Integrating this concept into step 3, we focus on the Lorentz-like adaptation for moving monopoles as given in step 3.
05

Write the Complete Expression

Compile the steps above, leading to: \( \mathbf{F} = q_m \mathbf{B} - q_m\mathbf{v} \times \mathbf{E} \). The first term involves the force directly from the magnetic field and the second term is due to the cross-product with the electric field.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lorentz Force Law
The Lorentz force law describes how an electric charge experiences a force when subjected to electric and magnetic fields. This principle is fundamental in electromagnetism and is represented by the equation: \[\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})\] where:
  • \( \mathbf{F} \) is the force on the particle,
  • \( q \) is the electric charge,
  • \( \mathbf{E} \) represents the electric field,
  • \( \mathbf{v} \) is the velocity of the charge, and
  • \( \mathbf{B} \) is the magnetic field.
The term \( \mathbf{E} \) indicates the force due to the electric field, while \( \mathbf{v} \times \mathbf{B} \) pertains to the magnetic field's influence, dependent on the velocity of the charge. For a magnetic monopole, this concept is adapted by swapping the roles of \( \mathbf{E} \) and \( \mathbf{B} \), thus introducing a new form of the Lorentz force for magnetic charges.
Coulomb's Law
Coulomb's law, originally formulated for electric charges, quantifies the force between two charged objects. In its traditional form, it states that the force between two electric charges, \( q_1 \) and \( q_2 \), is proportional to the product of their magnitudes, inversely proportional to the square of the separation distance, \( r \), and acts along the line connecting them: \[ \mathbf{F} = k \frac{q_1 q_2}{r^2} \hat{r} \] where \( k \) is Coulomb's constant and \( \hat{r} \) is the unit vector from one charge to the other. Analogously for magnetic monopoles, this principle is adjusted to the formula: \[ \mathbf{F} = \frac{\mu_{0}}{4 \pi} \frac{q_{m_1} q_{m_2}}{r^2} \mathbf{\varepsilon_1} \] where:
  • \( q_{m_1} \) and \( q_{m_2} \) are the magnetic charges,
  • \( \mu_0 \) represents the magnetic permeability of free space, and
  • \( \mathbf{\varepsilon_1} \) is a directional unit vector.
By understanding Coulomb's law, we can extend the interactions of magnetic monopoles similarly, noting the distinct variable roles for magnetic forces.
Electric and Magnetic Fields
Electric and magnetic fields are key components of electromagnetic theory describing forces acting at a distance.
  • Electric Field (\( \mathbf{E} \)): It represents a field surrounding an electric charge where another charge experiences a force. This field causes charges to experience a force in the direction of the field.
  • Magnetic Field (\( \mathbf{B} \)): It arises around magnets or due to moving electric charges and affects the motion of charged particles in its vicinity. The direction and magnitude of the field dictate the force's trajectory on a moving charge.
Both fields are interlinked through Maxwell's equations, and they can transform into one another under relative motion, a key insight of relativity theory. However, their effects on particles differ:- The electric field's force direction aligns with the field,- The magnetic field applies a perpendicular force, which can alter the particle's direction and speed depending on its velocity.Understanding these fields allows us to predict and manipulate the behavior of charges and monopoles in varying scenarios of movement and interaction.

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Most popular questions from this chapter

A familiar demonstration of superconductivity (Prob. 7.44) is the levitation of a magnet over a piece of superconducting material. This phenomenon can be analyzed using the method of images. \({ }^{31}\) Treat the magnet as a perfect dipole \(\mathbf{m}\), a height \(z\) above the origin (and constrained to point in the \(z\) direction), and pretend that the superconductor occupies the entire half-space below the \(x y\) plane. Because of the Meissner effect, \(\mathbf{B}=\mathbf{0}\) for \(z \leq 0\), and since \(\mathbf{B}\) is divergenceless, the normal (z) component is continuous, so \(B_{z}=0\) just above the surface. This boundary condition is met by the image configuration in which an identical dipole is placed at \(-z\), as a stand-in for the superconductor; the two arrangements therefore produce the same magnetic field in the region \(z>0\). (a) Which way should the image dipole point \((+z\) or \(-z) ?\) (b) Find the force on the magnet due to the induced currents in the superconductor (which is to say, the force due to the image dipole). Set it equal to \(M g\) (where \(M\) is the mass of the magnet) to determine the height \(h\) at which the magnet will "float" [Hint: Refer to Prob, 6.3.] (c) The induced current on the surface of the superconductor (the \(x y\) plane) can be determined from the boundary condition on the tangential component of \(\mathbf{B}\) (Eq. \(5.76\) ): \(\mathbf{B}=\mu_{0}(\mathbf{K} \times \hat{\mathbf{z}})\). Using the field you get from the image configuration, show that $$ \mathbf{K}=-\frac{3 m r h}{2 \pi\left(r^{2}+h^{2}\right)^{5 / 2}} \hat{\phi} $$ where \(r\) is the distance from the origin.

A long solenoid with radius \(a\) and \(n\) turns per unit length carries a time- dependent current \(I(t)\) in the \(\phi\) direction. Find the electric field (magnitude and direction) at a distance \(s\) from the axis (both inside and outside the solenoid), in the quasistatic approximation.

The magnetic field of an infinite straight wire carrying a steady current \(I\) can be obtained from the displacement current term in the Amp猫re/Maxwell law, as follows: Picture the current as consisting of a uniform line charge \(\lambda\) mov. ing along the \(z\) axis at speed \(v\) (so that \(I=\lambda v\) ), with a tiny gap of length \(\epsilon\), which reaches the origin at time \(t=0\). In the next instant (up to \(t=\epsilon / v\) ) there is no real current passing through a circular Amperian loop in the \(x y\) plane, but there is a displacement current, due to the "missing" charge in the gap. (a) Use Coulomb's law to calculate the z component of the clectric field, for points in the \(x y\) plane a distance \(s\) from the origin, due to a segment of wire with uniform density \(-\lambda\) extending from \(z_{1}=v t-\epsilon\) to \(z_{2}=v t\). (b) Determine the flux of this electric field through a circle of radius \(a\) in the \(x y\) plane. (c) Find the displacement current through this circle. Show that \(I_{d}\) is equal to \(I\), in the limit as the gap width \((\epsilon)\) goes to zero. 35

Two coils are wrapped around a cylindrical form in such a way that the same fluc passes through every turn of both coils. (In practice this is achieved by inserting an iron core through the cylinder; this has the effect of concentrating the flux.) The primary coil has \(N_{1}\) tums and the secondary has \(N_{2}\) (Fig. 7.57). If the current \(I\) in the primary is changing, show that the emf in the secondary is given by $$ \frac{\mathcal{E}_{2}}{\mathcal{E}_{1}}=\frac{N_{2}}{N_{1}} $$ where \(\mathcal{E}_{1}\) is the (back) emf of the primary. [This is a primitive transformer-a device for raising or lowering the emf of an alternating current source. By choosing the appropriate number of turns, any desired secondary emf can be obtained. If you think this violates the conservation of energy, study Prob. 7.58.]

Imagine a uniform magnetic field, pointing in the z direction and filling all space \(\left(B=B_{0} \hat{z}\right)\). A positive charge is at rest, at the origin. Now somebody turns off the magnetic field, thereby inducing an electric field. In what direction does the charge move? \(^{16}\)
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