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Section 4: Electromagnetism I

1. Coulomb's Law

Four point charges of +1.0 C each are placed at the corners of a square with sides of 1.0 m. Calculate the magnitude and direction of the electric force on a charge of -2.0 C placed at the center of the square.

2. Electric Potential

Point charges of +1C, -2C, +3C, and -4C are placed at the corners of a square with sides of 1.0 m (in order). Calculate the electric potential at the center of the square.

3. Electrostatic Equilibrium

Find the equilibrium position for a charge \(q_3 = +1\text{C}\) placed on the line between a charge \(q_1 = +4\text{C}\) and a charge \(q_2 = +9\text{C}\), which are separated by a distance of 2 m.

4. Force Comparison

Calculate the magnitude of the electric force and the gravitational force between an electron and a proton in a hydrogen atom (average distance \(r \approx 5.3 \times 10^{-11} \text{ m}\)). What is the ratio \(F_e/F_g\)?

5. Field Levitation

What electric field strength is required to make a proton levitate against Earth's gravity at the level of the Earth's surface? (Mass of proton \(m_p \approx 1.67 \times 10^{-27} \text{ kg}\), charge of proton \(e \approx 1.6 \times 10^{-19} \text{ C}\), gravitational acceleration \(g \approx 9.8 \text{ m/s}^2\)).

6. Field at a point from a system of charges

Two point charges are given:

  • \(+q\ \text{at point}\ (-a, 0)\)

  • \(+2q\ \text{at point}\ (a, 0)\)

  • Determine the field vector \(\vec E(0, y)\), \(\vec E(x, 0)\) and generally \(\vec E(x, y)\).

  • Determine the condition for which the components \(E_x = 0\), \(E_y = 0\) and the zero field \(\vec E = 0\).
  • Calculate the field for: \(a = 0.2\,\mathrm{m}\), \(y = 0.3\,\mathrm{m}\), \(q = 2\,\mu\mathrm{C}\).
  • Investigate the limit \(y \gg a\).

7. Cyclotron Motion

An electron is accelerated from rest through a potential difference of 5000 V. It then enters a region of uniform magnetic field B = 0.1 T, perpendicular to its velocity. What is the radius of the circular path it will follow?

8. Lorentz Force

A charged particle with charge \(q = 2 \times 10^{-19}\) C and mass \(m = 4 \times 10^{-27}\) kg enters a magnetic field of \(B = 0.5\) T at a speed of \(v = 10^6\) m/s perpendicular to the field. What is the magnitude of the Lorentz force acting on the particle?

9. Vector Lorentz Force

A proton moves with a velocity \(\vec{v} = (2\hat{i} - 4\hat{j} + \hat{k}) \text{ m/s}\) in a region where the magnetic field is \(\vec{B} = (\hat{i} + 2\hat{j} - \hat{k}) \text{ T}\). What is the magnitude of the magnetic force this charge experiences?

10. Lorentz Force acting on Wire

A straight wire 2.0 m long carries a current of 10 A. It is placed in a uniform magnetic field of B = 0.5 T. Calculate the magnetic force on the wire if the angle between the wire and the magnetic field is:

a) \(90^\circ\) b) \(45^\circ\) c) \(0^\circ\)