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Basics of Electromagnetic Fields

# Basics of Electromagnetic Fields

Electromagnetic fields (EMFs) are a fundamental aspect of physics, understood as physical fields produced by electrically charged objects. They affect the behavior of charged objects in the vicinity of the field. Electromagnetic fields extend indefinitely throughout space and can be seen as a combination of electric fields and magnetic fields.

## Electric Fields

An electric field is created by a charged object in the space surrounding it, and its effect is to exert a force on other charged objects in the field. The strength of an electric field is measured in volts per meter (V/m). The direction of the field is taken as the direction that a positive test charge would be pushed when placed in the field.

The electric field \(\vec{E}\) due to a point charge \(Q\) at a distance \(r\) is described by Coulomb’s law:

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\[
\vec{E} = k_e \frac{Q}{r^2} \hat{r}
\]

where \(k_e\) is Coulomb’s constant (\(8.9875 \times 10^9 Nm^2/C^2\)), \(Q\) is the charge, \(r\) is the radius/distance from the charge, and \(\hat{r}\) is the unit vector in the direction from the charge.

## Magnetic Fields

A magnetic field is a field of force created by moving charges (current) and magnetic dipoles, and has both a magnitude and a direction. The strength of a magnetic field is measured in teslas (T) or gauss (G), where 1 T = 10,000 G.

The magnetic field \(\vec{B}\) around a long, straight conductor with current \(I\) can be calculated with Ampere’s Law:

\[
\vec{B} = \frac{\mu_0 I}{2\pi r} \hat{\phi}
\]

where \(\mu_0\) is the permeability of free space (\(4\pi \times 10^{-7} Tm/A\)), \(I\) is the current, \(r\) is the distance from the wire, and \(\hat{\phi}\) is the unit vector in the direction of the magnetic field.

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## Electromagnetic Induction

Electromagnetic induction occurs when a time-varying magnetic field creates an electric field. This is described by Faraday’s law, which states that the induced electromotive force (EMF) in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit.

Faraday’s law can be written as:

\[
\text{EMF} = -\frac{d\Phi_B}{dt}
\]

where \(\Phi_B \) is the magnetic flux through the circuit.

## Maxwell’s Equations

Maxwell’s equations are the set of four fundamental equations that describe how electric fields and magnetic fields propagate and how they are generated by charges and currents.

1. Gauss’s law for electricity (electric flux):
\[
\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}
\]

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2. Gauss’s law for magnetism (magnetic field lines are closed loops):
\[
\oint \vec{B} \cdot d\vec{A} = 0
\]

3. Faraday’s law of induction (EMF induced by time-varying magnetic field):
\[
\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}
\]

4. Ampère’s Circuital Law (magnetic field around a conductor):
\[
\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} + \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt}
\]

# Problems and Solutions

Here are 20 problems with solutions related to the basics of electromagnetic fields.

Problem 1: Calculate the electric field intensity at a distance of 1 meter from a point charge of \(2 \times 10^{-6} C\).
Solution:
\[
\vec{E} = k_e \frac{Q}{r^2} = (8.9875 \times 10^9 Nm^2/C^2) \frac{2 \times 10^{-6} C}{(1 m)^2} = 17975 \, \text{N/C}
\]

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