Derive the expression (7.27) for the magnetic B-field due to a moving charge by using the displacement current.

The Biot-Savart Law is an equation that describes the magnetic field generated by a steady electric current. It's fundamental in understanding how moving charges create magnetic fields. According to its formulation, the magnetic field \textbf{B} due to an element of current can be described as:

\textbf{B} = \frac{\mu_0}{4\pi} \frac{Id\textbf{s} \times \hat{\textbf{r}}}{r^2},

where \( \mu_0 \) is the permeability of free space, \( I \) is the current, \( d\textbf{s} \) is the differential element of the current's path, \( \hat{\textbf{r}} \) is the unit vector pointing from the current element towards the observation point, and \( r \) is the distance from the current element to the observation point. For a moving point charge, the equivalent form of the Biot-Savart Law is used, as seen in the exercise solution, where the current element is replaced with a moving charge producing a magnetic field in the surrounding space.

Displacement current plays a crucial role in connecting electricity and magnetism. It was introduced by Maxwell to adjust the conditions under which a time-varying electric field could produce a magnetic field, similar to a physical current. The term displacement current refers to a quantity \textbf{I}_d determined by the changing electric field in a given region. It is given by:

I_d = \varepsilon_0 \frac{d\Phi_E}{dt},

where \( \varepsilon_0 \) is the permittivity of free space and \( \Phi_E \) is the electric flux through a surface. Although it does not correspond to actual charges moving in space, displacement current is treated as a source of the magnetic field in Maxwell's equations, thereby closing the gap that previously existed in Ampère's circuital law for situations where no physical current is present.

Maxwell's equations are a set of four fundamental equations that form the foundation of classical electromagnetism. They describe how electric and magnetic fields are generated and altered by each other and by charges and currents. These equations not only predict the existence of electromagnetic waves but also tie together the electric and magnetic fields into a coherent framework. Maxwell's equations consist of:

• Gauss's law for electricity
• Gauss's law for magnetism
• Faraday's law of induction
• Maxwell-Ampère's law (modified Ampère's circuital law)

Gauss's law describes the electric field arising from charges, Gauss's law for magnetism states that there are no magnetic monopoles, Faraday's law connects changing magnetic fields with induced electric fields, and the modified Ampère's law includes displacement current as a source of the magnetic field.

The modified Ampère's circuital law includes both conduction current and the displacement current as sources of the magnetic field. It provides a complete description of the magnetic field around a conductor carrying a steady current as well as in areas with a time-varying electric field. The law can be expressed as:

\oint \textbf{B} \cdot d\textbf{l} = \mu_0 (I_c + I_d),

where \( \oint \textbf{B} \cdot d\textbf{l} \) represents the line integral of the magnetic field \textbf{B} around a closed loop, \( \mu_0 \) is the permeability of free space, \( I_c \) the conduction current passing through the loop, and \( I_d \) the displacement current through the same area. This modified version of Ampère's law accommodates changing electric fields, as demonstrated in the step-by-step solution for the magnetic field due to a moving charge—offering students a more accurate insight into the dynamic relationship between electricity and magnetism.