Turn In Your Answers To The Following Problems 1 Rev 911 Two

Turn In Your Answers To The Following Problems1 Rev 911 Two Alph

Identify the core physics problems involving electrostatics and magnetism, and provide detailed solutions with appropriate physics formulas, calculations, and explanations. The specific questions include calculating electrostatic force between two charges, determining medium permittivity based on electron-electron force, analyzing magnetic field directions using the right-hand rule, calculating magnetic field strength around a current-carrying wire, and assessing the Lorentz force on a wire in different magnetic field orientations.

Paper For Above instruction

Understanding the fundamental principles of electrostatics and magnetism is vital in physics, especially when analyzing forces and fields involving electric charges and currents. This paper addresses select problems that exemplify the calculation of electrostatic forces, the influence of mediums on these forces, the application of the right-hand rule to magnetic forces, and the determination of magnetic fields around conductors, along with the resulting Lorentz forces.

1. Electrostatic Force Between Two Alpha Particles

Given two alpha particles, each with a charge of \(q = 3.20 \times 10^{-19}\) C, separated by a distance of 1 Å (which equals \(1 \times 10^{-10}\) meters). To calculate the electrostatic force between them, we apply Coulomb's Law:

\[ F = \frac{k \, |q_1 q_2|}{r^2} \]

where \(k\) is Coulomb's constant (\(8.988 \times 10^9 \, \mathrm{Nm^2/C^2}\)). Substituting the given values:

\[ F = \frac{(8.988 \times 10^9) \times (3.20 \times 10^{-19})^2}{(1 \times 10^{-10})^2} \]

Simplifying numerator and denominator:

\[ F = \frac{8.988 \times 10^9 \times 10.24 \times 10^{-38}}{1 \times 10^{-20}} \]

\[ F = 8.988 \times 10^9 \times 10.24 \times 10^{-18} \]

\[ F \approx 92.09 \times 10^{-9} \text{ N} \]

or approximately 92.1 nanonewtons.

As both particles have like charges, the force is repulsive.

2. Permittivity of the Medium from Electron-Electron Force

Given two electrons with charge \(-1.602 \times 10^{-19}\) C, separated by 1 Å, experiencing a force of \(3.8 \times 10^{-9}\) N. To find the permittivity \(\varepsilon\) of the medium, Coulomb's law in a medium is expressed as:

\[ F = \frac{1}{4\pi \varepsilon} \frac{|q_1 q_2|}{r^2} \]

Rearranged to solve for \(\varepsilon\):

\[ \varepsilon = \frac{1}{4 \pi} \frac{|q_1 q_2|}{F r^2} \]

Substituting known values:

\[ \varepsilon = \frac{1}{4 \pi} \times \frac{(1.602 \times 10^{-19})^2}{(3.8 \times 10^{-9}) \times (1 \times 10^{-10})^2} \]

Calculating numerator:

\[ (1.602 \times 10^{-19})^2 = 2.566 \times 10^{-38} \]

And denominator:

\[ 3.8 \times 10^{-9} \times 1 \times 10^{-20} = 3.8 \times 10^{-29} \]

Thus:

\[ \varepsilon = \frac{1}{4 \pi} \times \frac{2.566 \times 10^{-38}}{3.8 \times 10^{-29}} \]

\[ \varepsilon = \frac{1}{12.566} \times 6.75 \times 10^{-10} \]

\[ \varepsilon \approx 5.38 \times 10^{-11} \, \mathrm{F/m} \]

This value is close to the permittivity of free space (\(\varepsilon_0 = 8.854 \times 10^{-12} \mathrm{F/m}\)), indicating the medium is similar to vacuum or a low-permittivity medium.

3. Magnetic Force Direction and the Right-Hand Rule

Considering an electron moving in the x-y plane experiencing a force in the z-direction due to a magnetic field, the Lorentz force law states:

\[ \vec{F} = q \, \vec{v} \times \vec{B} \]

Since the force is vertical (either up or down), the magnetic field must be oriented such that the cross product of the electron's velocity vector and the magnetic field vector results in a force in the z-direction.

If the electron moves in the x-direction (\(\vec{v}\) along x), and the force is upward (\(+z\)), then using the right-hand rule: point your fingers in the direction of \(\vec{v}\) (x-axis), curl toward \(\vec{B}\), and your thumb points in the direction of \(\vec{F}\). To get an upward force, the magnetic field must be directed along the y-axis. Conversely, a downward force implies the magnetic field is opposite in direction.

Thus, the possible magnetic field directions are along the y-axis, either positive or negative, depending on the force's direction. This demonstrates how the right-hand rule helps determine the magnetic field orientation for forces acting perpendicular to the velocity of charged particles.

4. Magnetic Field Around a Long Straight Wire

Using Ampère's Law, the magnetic field \(B\) at a distance \(r\) from a long, straight wire carrying current \(I\) is given by:

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

Where \(\mu_0 = 4\pi \times 10^{-7} \, \mathrm{T\,m/A}\), \(I=100\,\mathrm{A}\), and \(r=1\,\mathrm{cm} = 0.01\,\mathrm{m}\). Substituting the values:

\[ B = \frac{4\pi \times 10^{-7} \times 100}{2 \pi \times 0.01} \]

Simplify numerator and denominator:

\[ B = \frac{4\pi \times 10^{-5}}{2 \pi \times 0.01} \]

\[ B = \frac{4 \times 10^{-5}}{2 \times 0.01} = \frac{4 \times 10^{-5}}{0.02} \]

\[ B = 2 \times 10^{-3} \, \mathrm{T} \]

So, the magnetic field strength one centimeter away from the wire is 2 milliteslas (or 0.002 T).

5. Lorentz Force on a Current-Carrying Wire

Given a wire carrying a current \(I = 10\,\mathrm{mA} = 1 \times 10^{-2}\, \mathrm{A}\) immersed in a magnetic field \(B=5\, \mathrm{T}\). The Lorentz force per unit length (force per meter) on a current-carrying wire in a magnetic field is:

\[ \frac{F}{L} = I \times B \times \sin \theta \]

where \(\theta\) is the angle between the wire and magnetic field lines.

Case 1: Magnetic field lines are perpendicular (\(\theta=90^\circ\)).

\[ \frac{F}{L} = (1 \times 10^{-2}) \times 5 \times 1 = 5 \times 10^{-2} \, \mathrm{N/m} \]

Case 2: Magnetic field lines are parallel to the wire (\(\theta=0^\circ\)).

\[ \frac{F}{L} = (1 \times 10^{-2}) \times 5 \times 0 = 0 \, \mathrm{N/m} \]

Thus, the magnetic force per meter is 0.05 N/m when the magnetic field is perpendicular to the wire, and zero when parallel, confirming the importance of orientation in magnetic force interactions.

Conclusion

These analyses underscore core electromagnetic principles: Coulomb's Law for electrostatic forces, the impact of media via permittivity, the directional nature of magnetic forces with the right-hand rule, the calculation of magnetic fields around conductors, and the dependence of Lorentz force on the orientation between current and magnetic fields. Such calculations and conceptual understandings form the foundation of electromagnetism with practical applications spanning electronics, magnetic resonance, and electromagnetic shielding.

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