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What is the direction of the electric field at the centre of a uniformly charged circular ring?
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What is the total capacitance of 10 capacitors, each of capacitance 1 µF, connected in parallel?
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What is the potential difference between points A and B in the given circuit?
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A loop carrying a current I clockwise is placed in x-y plane, in a uniform magnetic field directed along z-axis. What is the direction of the magnetic force on the loop?
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What is the magnitude of the electric field at the centre of a uniformly charged circular ring of radius R and charge density λ?
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Ten capacitors, each of capacitance 1 µF, are connected in parallel to a source of 100 V. What is the total energy stored in the system?
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A thin plastic rod is bent into a circular ring of radius R and uniformly charged with charge density λ. What is the magnitude of the electric field at its centre?
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What is the direction of the magnetic field at the centre of a circular loop carrying a current I?
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A loop carrying a current I counterclockwise is placed in x-y plane, in a uniform magnetic field directed along z-axis. What is the direction of the magnetic force on the loop?
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What is the total capacitance of 10 capacitors, each of capacitance 1 µF, connected in series?
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Study Notes
Electric Dipole and Potential Energy
- An expression for the potential energy ( U ) of an electric dipole ( \mathbf{p} ) in a uniform electric field ( \mathbf{E} ) is given by ( U = -\mathbf{p} \cdot \mathbf{E} ).
- Potential energy is maximum when the dipole is aligned against the electric field and minimum when aligned with it.
- Example: An electric dipole consisting of point charges ( 1.0 , \text{pC} ) and ( -1.0 , \text{pC} ) located at points (0, 0) and (3 mm, 4 mm) can be analyzed in a ( 1000 , V/m ) electric field for torque calculation.
Electric Dipole Potential
- For an electric dipole formed by charges ( q ) separated by distance ( 2a ) along the x-axis, the potential ( V ) at a point ( x ) (where ( x \gg a )) is expressed as ( V = \frac{1}{4\pi \epsilon_0}\frac{\mathbf{p} \cdot \hat{r}}{r^2} ) where ( \mathbf{p} ) is the dipole moment.
- Metallic spheres S1 and S2 (radii 1 cm and 3 cm respectively) with the same charge density ( 10^9 , C/m^2 ) can redistribute charge through a connecting wire, leading to a change in charge on sphere S1.
AC Circuits and Impedance
- When a resistor ( R ) and capacitor ( C ) are connected in series to an AC source ( v = V_m \sin(\omega t) ), the impedance ( Z ) can be derived considering resistance and reactance, given by ( Z = \sqrt{R^2 + (X_C)^2} ).
- Inductors appear as conductors in alternating current circuits under certain conditions, mainly determined by frequency and inductance.
Gauss's Law and Electric Flux
- For a cube of side 0.1 m in an electric field ( 500 , N/C ) directed along the x-axis, the electric flux through the cube and the enclosed charge can be calculated using Gauss's Law: ( \Phi_E = E \cdot A ).
Current Density in Conductors
- Current density ( \mathbf{j} ) is defined as the electric current per unit area, treated as a vector quantity.
- The relationship ( \mathbf{j} = n e \mathbf{E} ) emerges in conductors, where ( n ) is the number of charge carriers per unit volume and ( e ) is the charge of an electron.
Capacitors and Energy Storage
- The total energy stored in a parallel combination of capacitors is calculated using the formula ( U = \frac{1}{2}CV^2 ), where ( C ) is the total capacitance and ( V ) the voltage across the capacitors.
Loop in Magnetic Field
- A current-carrying loop placed in a uniform magnetic field experiences a torque, influenced by the plane of the loop and the direction of the magnetic field.
Circuit Analysis and Potential Difference
- Analyzing simple circuits requires understanding the principles of voltage division and Kirchhoff's laws to determine potential differences between points in the circuit.
Electric Dipole and Potential Energy
- Potential energy ( U ) of an electric dipole in a uniform electric field ( E ) is given by the formula:
( U = -\mathbf{p} \cdot \mathbf{E} ), where ( \mathbf{p} ) is the dipole moment. - Maximum potential energy occurs when the dipole is aligned opposite to the electric field (( \theta = 180^\circ )).
- Minimum potential energy occurs when the dipole is aligned with the electric field (( \theta = 0^\circ )).
- An electric dipole made of point charges ( -1.0 , \text{pC} ) and ( +1.0 , \text{pC} ) is located at points ( (0, 0) ) and ( (3 , \text{mm}, 4 , \text{mm}) ) in an electric field ( E = 1000 , \hat{i} , \text{V/m} ).
- Torque ( \tau ) acting on the dipole is calculated using the formula ( \tau = \mathbf{p} \times \mathbf{E} ).
Potential of an Electric Dipole
- When an electric dipole moment ( \mathbf{p} = p \hat{i} ) is oriented along the x-axis with charges ( q ) separated by distance ( 2a ), the potential ( V ) at a point ( x ) far away (( x >> a )) is approximately:
( V = \frac{1}{4\pi \epsilon_0} \cdot \frac{\mathbf{p} \cdot \hat{i}}{x^2} ).
Charge Density and Spheres
- Two isolated metallic spheres ( S_1 ) (radius ( 1 , \text{cm} )) and ( S_2 ) (radius ( 3 , \text{cm} )) are charged to have the same charge density of ( 10^9 , \text{C/m}^2 ).
- When connected by a wire, the new charge on sphere ( S_1 ) can be calculated using the formula for charge density and surface area.
Series Circuit with Resistor and Capacitor
- Impedance ( Z ) in an AC circuit with a resistor and capacitor in series can be derived showing ( Z = \sqrt{R^2 + \left(\frac{1}{\omega C}\right)^2} ).
- An inductor acts as a conductor when its reactance is negligible compared to resistance or at certain resonant frequencies.
Electric Flux and Charge
- An electric cube with side ( 0.1 , \text{m} ) placed in an electric field ( E = 500 \hat{i} , \text{N/C} ) requires calculating electric flux and enclosed charge using Gauss's law.
- Electric flux ( \Phi = E \cdot A ), where ( A ) is the area of the cube face perpendicular to the electric field.
- The enclosed charge ( Q ) can be determined from ( \Phi = \frac{Q}{\epsilon_0} ).
Current Density in Conductors
- Current density ( j ) is defined as the electric current per unit cross-sectional area, making it a vector quantity.
- In a metallic conductor, given the number of electrons per unit volume ( n ) and relaxation time ( \tau ), the relationship is expressed as:
( j = n e \mathbf{E} ), where ( e ) is the charge of an electron.
Capacitor Energy Storage
- For ten capacitors, each with capacitance ( 1 , \mu\text{F} ) connected in parallel to a ( 100 , \text{V} ) source, total energy stored ( U ) is given by:
( U = \frac{1}{2} C V^2 ). The value is determined based on combined capacitance and voltage.
Loop and Magnetic Field
- A current-carrying loop in a uniform magnetic field experiences torque, depending on the orientation of the current with respect to the field direction. The behavior under different configurations should be analyzed.
These notes encompass critical aspects of electric dipoles, charge interactions, and basic principles of electric circuits and fields, providing a solid foundation for further study in electromagnetism.
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