Consider a 4 nF parallel-plate capacitor with nothing between the plates that is charged by a battery at a potential difference of V. The capacitor stores an energy of 1.02 × 10^-5 J. The volume between the plates is totally filled with a sheet of dielectric while the power source is still connected. As a result, the energy-storing capacity of the capacitor is enhanced by 2.78 × 10^-5 J. i) Determine V and ii) the dielectric constant (K) of the sheet.

Consider a cylindrical capacitor composed of two concentric conducting cylinders with inner and outer radii, R_in and R_out. A strong, nonuniform electric field is created in the space between these cylinders, designed to ionize the air in this region. When the electric field strength exceeds the dielectric breakdown threshold of air (E_breakdown = 3.0 x 10⁶ N/C), air molecules in the region become ionized, resulting in free electrons and positively charged ions.

This ionization occurs in a specific cylindrical region around the inner conductor, extending to a radius r, which is typically five times the radius of the inner cylinder. In this setup, the inner radius of the cylinder is R_in = 0.15 mm, and the outer radius is R_out = 12.0 cm.

To generate the required electric field strength for ionization at a radius of r = 5.0 R_in, what potential difference V should be applied between the inner and outer conducting cylinders?

A Scanning Electron Microscope generates an electron beam that carries a current of 60 μA. The beam scans the surface of a sample, allowing the user to see minute details on the specimen. How many electrons strike the sample every second?

A thermistor has a resistance that varies with temperature according to the equation R(T) = R  ₀ e^{[C ((1/T) - (1/298 K))]}. Here, R₀ is the thermistor resistance at 298 K, T is the temperature in Kelvin, and C is a constant with units of Kelvin. The thermistor is connected to a 12.0 V battery. At a temperature of 25.0°C, the current flowing through the thermistor is 3.50 mA. When the temperature of the thermistor increases to 40.0°C, the current flowing through it increases by 10%. Determine the constant C.

A vehicle has two 12.0 V batteries connected in series. The combined batteries drive a starter motor of resistance 5.40 Ω. If the batteries are ideal, determine the power supplied to the motor.

Determine the drift speed of the electron, given that 2.1 × 10¹⁹ electrons pass through a cross-section of a 1.4 mm diameter copper wire in 12s. The number density of electrons in copper is 8.5 × 10²⁸ m^-3.

A room heater has two heating coils connected in parallel. Their resistance is R₁ = 14.4 Ω and R₂ = 10.3 Ω and the power line provides a 110 V potential difference. Determine the current through each coil.

An experimental setup consists of a battery with an EMF of 6.0 V and three resistors as shown below. Calculate the current passing through the battery and V_A- V_Bwhen the switch is (a) open and b) closed.

A capacitor of 15 μF capacitor is discharged through an unknown resistor. The current flowing through the resistor is measured using an ammeter. The recorded data is presented in the table below. Determine:

i) the resistance

ii) the initial voltage across the capacitor.

Consider a circuit consisting of a 9.0-V battery, a 100.0 μF capacitor, a 60.0 Ω resistor, a switch, and an ammeter wired in series. Initially, the switch is open and the capacitor is totally discharged. The switch is closed, and after a time t, the ammeter shows a value of 0.05 A. Determine the charge Q on the capacitor at time t.

A source generates a uniform magnetic field of strength 1.15 T that is directed at an angle of 37.0° from the z-direction. A water pitcher made of glass with a circular plastic top lid of 7.00 cm radius lies, within the magnetic field, on the top of a horizontal table. Determine the total magnetic flux passing through the pitcher glass.

Determine the electric and magnetic fields at the surface of a cylindrical conductor with diameter d, length = 10d, and conductivity σ, when there is current I passing through it. Assume a uniform electric field throughout, including at the surface of the conductor.

A proton with a velocity v = 3.6 × 10⁶ ĵ m/s is shot into an electric field E = (3.0 × 10⁵ î + 3.0 × 10⁵ ĵ) V/m, and B = 0.20 k̂ T. Determine the total force experienced by the proton.

A beam of singly charged particles of ⁷Li moves through a uniform magnetic field at a speed of 4.0 km/s. A ⁷Li particle has a mass of 1.16 × 10^-26 kg. The magnetic field deflects the particles' beam, and the beam leaves the magnetic field region perpendicular to the initial direction of incidence. Determine the strength of the magnetic field if the distance traveled by the ⁷Li beam is 3.24 cm.

Positively charged alpha particles move in a circular path with a radius of about 4.0 cm under the influence of a magnetic field with a strength equal to about 1.0 T. What magnitude and direction for an applied electrical field would cause these alpha particles' trajectory to become linear? [Hint: The mass of an alpha particle = 6.644 × 10^-27 kg, the charge of an alpha particle = +2 ×1.602 × 10^-19 C.]

A segment of a long current-carrying conductor is placed within a uniform 0.340 T magnetic field, pointing directly out of the plane of the page. The conductor makes two right-angle turns and is positioned as shown in the figure. A current of I = 5.00 A flows within the conductor. What are i) the magnitude and ii) the direction of the magnetic force acting on the conductor due to the magnetic field?

What is the maximum torque on a 18-turn square coil (4.0 cm per side) that spins in a 0.030 T magnetic field, if a motor powered by a 8.0-V battery has a total resistance of 22 Ω?

In a laboratory experiment, a solenoid made of 425 turns of copper wire arranged in a cylindrical shape is used. The length and the diameter of the solenoid are 32.0 cm and 4.0 cm, respectively. A current of 5.0 A flows through the copper wire. Determine the magnetic field at the center of the solenoid.

The electric current flowing through an element in a circuit follows the relation 30A - (0.25 A/s²)t². Determine the coulomb of charge that passes a fixed point in the element within 12.0 s.

A current-carrying wire is placed as depicted in the figure below. Determine the magnetic field vectors at points X, Y, and Z. Provide your answers as vectors, indicating both magnitude and direction.

Two long electric lines are suspended vertically. Line 1 carries a 2.20 A downward current. An upward current of 3.20 A flows through line 2, which is 8.0 cm to the left of line 1. Find the magnitude of the force exerted on a 0.5 m length of one line by the other line. What is the nature of force between two parallel lines?

Aiming to determine how the magnetic field of a perfect toroid having 300 turns varies with distance from the torus center, an electronic circuit engineer sends a 6.00 A current into the toroidal coil. The toroid's internal and external diameters are 16.0 cm and 26.0 cm, respectively. Find the magnetic field magnitude measured by the engineer at i) 6.0 cm, ii) 10.0 cm, and iii) 15.0 cm from the toroid's central axis.

A conducting loop carries a steady, counterclockwise current I. The loop consists of two half-circles connected by vertical, equal-length straight sections, forming a horizontal loop. The radius of the right half-circle is R_A (smaller radius) and the radius of the left half-circle is R_B (larger radius).

(i) Find the magnitude and direction of the magnetic field at the center point P of the loop. (ii) Calculate the magnetic dipole moment of the loop.

Calculate the line integral of B̂•dŝ between points A and B.