Physics Concept Notes for JEE: Electromagnetic Induction Magnetism and Capacitors
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Electromagnetic Induction
1. Faraday's law: the induced emf is \(e = -\dfrac{d\phi}{dt}\), where \(\phi = BA\cos\theta\).
2. Lenz's law: the induced current opposes the change of flux that causes it. The minus sign in Faraday's law expresses this.
3. Motional emf of a rod of length l moving with speed v across a field: \(e = Blv\), and it needs v, B and l to be mutually perpendicular.
4. Mutual inductance: \(e_2 = -M\dfrac{dI_1}{dt}\) and \(M_{12} = M_{21}\). Self inductance: \(e = -L\dfrac{dI}{dt}\).
5. At the instant a circuit with an inductor is switched on, the inductor carries no current and behaves like an open circuit. After a long time it acts as a plain wire.
6. Eddy currents are induced in bulk conductors by changing flux. They oppose the motion and heat the conductor, which damps a pendulum swinging through a field.
Direction rule for a straight wire: with the current upward, the field is out of the page to the left of the wire and into the page on its right. A loop on either side then gets an induced current that opposes the change of that flux.
Electromagnetic Waves
1. Waves are transverse. \(\vec{E}\), \(\vec{B}\) and the direction of travel are mutually perpendicular, and the wave travels along \(\vec{E}\times\vec{B}\).
2. In vacuum \(\dfrac{E_0}{B_0} = c\) and \(c = \dfrac{1}{\sqrt{\mu_0\varepsilon_0}}\).
3. Intensity \(I = \tfrac12\varepsilon_0 c E_0^2\). For a point source of power P, \(I = \dfrac{P}{4\pi r^2}\).
4. Peak and rms values are related by \(E_0 = \sqrt{2}\,E_{rms}\).
5. Displacement current \(I_d = \varepsilon_0\dfrac{d\Phi_E}{dt}\) comes from a changing electric field, with no flow of charge, and it makes Ampere's law consistent.
Moving Charges and Magnetism
| Topic | Result |
|---|---|
| Force on a charge | \(\vec{F} = q(\vec{v}\times\vec{B})\) |
| Radius and frequency of the circular path | \(r = \dfrac{mv}{qB} = \dfrac{\sqrt{2mK}}{qB}\), \(f = \dfrac{qB}{2\pi m}\) |
| Force on a wire of any shape in a uniform field | \(\vec{F} = I\,\vec{\ell}_{ends}\times\vec{B}\), using the straight line between the ends |
| Field at the centre of a coil | \(B = \dfrac{\mu_0 NI}{2r}\) |
| Long straight wire of radius a | \(B = \dfrac{\mu_0 I r}{2\pi a^2}\) inside, \(\dfrac{\mu_0 I}{2\pi r}\) outside |
| Force per unit length between parallel currents | \(\dfrac{\mu_0 I_1I_2}{2\pi d}\), attractive when currents are in the same direction |
| Torque on a coil | \(\tau = NIAB\sin\theta\), maximum when the field lies in the plane of the coil |
Orbital magnetic moment of a charge: \(\vec{M} = \dfrac{q}{2m}\vec{L}\). A bent magnet has moment equal to pole strength times the straight distance between the poles. Paramagnetic and ferromagnetic atoms have a permanent moment, diamagnetic atoms do not.
Capacitors
| Idea | Result |
|---|---|
| Parallel | \(C = C_1 + C_2\), same voltage |
| Series | \(\dfrac{1}{C} = \dfrac{1}{C_1} + \dfrac{1}{C_2}\), same charge, the smaller C takes the larger voltage |
| Energy stored | \(U = \tfrac12 CV^2 = \dfrac{Q^2}{2C}\) |
| Work done by a battery | \(QV\), twice the stored energy |
| Two charged capacitors joined | common potential = total charge divided by total capacitance |
| Plates pulled apart after disconnection | Q constant, C falls, U rises |
In the steady state a capacitor carries no current, so a resistor in series with it has no voltage drop.
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