Electricity And Magnetism Notes

Electricity And Magnetism Notes

An electric current is the rate of flow of electric charges.

In units, \(1 A=\frac{1 C}{1 s}\)

• • Average current, \(I_{\mathrm{av}}=\frac{\Delta Q}{\Delta t}\)
  • Instantaneous current, \(i=\frac{d Q}{d t}\)
• The current density is the current per unit cross-sectional area.

In SI units, \(1 \mathrm{Am}^{-2}=\frac{1 \mathrm{~A}}{1 \mathrm{~m}^2}\)

• • Average current density, \(\vec{j}=\frac{\Delta I}{\Delta s}\)
    • So,\(\Delta I=\vec{j} \cdot \overrightarrow{\Delta s}\)
  • Current density at some specific point, \(\vec{j}=\frac{d I}{d s}\)
    • So, \(I=\int \vec{j} \cdot \overrightarrow{d s}\)

Note that current is not a vector and is additive, but current density is a vector quantity.

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Drift speed, \(v_d=\frac{l}{n A e}\), where n = number density, e = charge of an electron and A = cross-sectional area.

• Ohm’s law: V = IR (in file scalar form).
  • \(\vec{j}=\sigma \vec{E}\)(in the vector form).
  • Resistance, \((\rho)=\frac{1}{\text { conductivity }(\sigma)}\)
• Resistance, \(R=\rho \frac{l}{A}\).
• Temperature-dependence of resistance: R_θ = R₀ (1+αθ), where a = temperature coefficient of resistance.
• Resistance in series: Rs = R1 + R2 +… where Rs is greater than the greatest resistance.
• Resistance in parallel: \(\frac{1}{R_p}=\frac{1}{R_1}+\frac{1}{R_2}+\ldots\)+…, where R is less than the least resistance.

The electromotive force (emf) ε of an electric cell is the potential difference between the terminals of the cell (or terminal voltage) in an open circuit.

• The terminal voltage of a cell of emf ε and internal resistance r:
  • When the cell delivers a current I,
    V = V_A-V_B = (V_A-V_C) + (V_C-V_B)
    = ε-(V_B-V_c)= ε-Ir  [∵V_B-V_c]

• • When the cell is being charged by a steady current I,
    V = V_A-V_B
    = (V_A-V_C) + (V_C-V_B)
    =ε+Ir

Magnetism and Matter NEET Notes

• The current through a shunt and a galvanometer are respectively

⇒ \(I_{\mathrm{s}}=\left(\frac{G}{G+S}\right) I \text { and } I_g=\left(\frac{S}{G+S}\right) I \text {, }\)

physics magnetism and electricity

where S = shunt resistance

and G = Galvanometer

• Galvanometer as an ammeter.
  • I_g G = (I-I_g)S.
  • \(S=\frac{G}{n-1}\)
  • \(n=\frac{I}{I_g}\)

• Galvanometer as a voltmeter:
  • V_A-V_B = I_g (R+G).
  • R = (n-1)G.
  • \(n=\left(\frac{V}{V_{\mathrm{g}}}\right)=\frac{V}{I_{\mathrm{g}} G}\)

Magnetism and Matter NEET Notes

Kirchhoff’s laws:

• • Junction rule: ∑I_i= 0 (= charge conservation),
  • Loop rule: ∑V_i = 0 (= energy conservation).

Grouping of cells:

• • In series: \(I=\frac{N e}{R+N r}.\)
  • In parallel: \(I=\frac{N e}{N R+r}\)
• Equivalent emf of two cells in parallel:

\(\mathcal{E}=\frac{e_1 r_2+e_2 r_1}{r_1+r_2}\), While the main current is

⇒ \(I=\frac{e_1 r_2+e_2 r_1}{R\left(r_1+r_2\right)+r_1 r_2}\).

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• Wheatstone bridge: The bridge shown in the figure is said to be balanced if \(\frac{P}{Q}=\frac{R}{S}\).

• Equivalent resistance in some special cases across A and B:

Magnetic Effect of Current formulae for NEET

\(R_{\mathrm{AB}}=\left(\frac{3 n+1}{n+3}\right) r\)

R_AB = r₁ + r₂

The CD is a conductor.

• Electric Power:

P (in Watts) = V (in Watts) x I (im amperes)= \(\frac{V^2}{R}=I^2 R\)

• Equivalent Power:
  • In series: \(\frac{1}{W_s}=\frac{1}{W_1}+\frac{1}{W_2}+\ldots\)
  • In parallel: WP = W1 + W2 +…
  • A Potentiometer measures the emf of a cell.
  • Potential gradient, \(k=\frac{V_{\mathrm{A}}-V_{\mathrm{B}}}{A B}\)
  • This internal resistance of the cell, \(r=\left(\frac{l}{l}-1\right) S\), where balancing lengths are l and l with key (K) open and closed respectively.

• Growth of change in an RC circuit:
  • Instantaneous charge, \(q=Q_0\left(1-e^{-t / R C}\right)\)
  • Then, instantaneous current, \(i=\frac{d q}{d t}=\frac{Q_0}{R C} e^{-t / R C}=\frac{\varepsilon}{R} e^{-t / R C}\)
• Decay of charge in an RC circuit:
  • Instantaneous charge, \(q=Q_0 e^{-t / R C}=\varepsilon C e^{-t / R C}\).
  • Then, instantaneous current, \(i=\frac{d q}{d t}=-\frac{\varepsilon}{R} e^{-t / R C}\).
  • The time constant of an RC circuit = RC (measured in seconds).

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Magnetic effect of current: Moving charges, (collectively equivalent to an an electric current) produce a magnetic field\((\vec{B})\) around themselves.

• Force on a charge q moving with a velocity \(\vec{v}\).
  • In an electric field \(\vec{E}: \quad \vec{F}_{\text {elec }}=q \vec{E}\).
  • In a magnetic field \(\vec{B}: \vec{F}_{\mathrm{mag}}=q(\vec{v} \times \vec{B})\).
• Lorentz force: \(\vec{F}=\vec{F}_{\text {elec }}+\vec{F}_{\text {mag }}=q(\vec{E}+\vec{v} \times \vec{B})\).
• Path of a charged particle moving in a uniform magnetic field \(\overrightarrow{\boldsymbol{B}}\):

The force \(\vec{F}=q(\vec{v} \times \vec{B})\) acting perpendicular to both \(\vec{v}\)and \(\vec{B}\) provides a centripetal force for its circular path of radius r, where \(F=q v B=\frac{m v^2}{r}, v=\frac{q B r}{m}\) (i.e v r)and time of revolution \(T=\frac{2 \pi m}{q B}\)(independent of radius r).

Magnetic Effect of Current formulae for NEET

• Force on a current element \((I \overrightarrow{d l})\)in a magnetic field \(\overrightarrow{\boldsymbol{B}}\):

\(\vec{F}=q \vec{v} \times \vec{B}=I(d t \vec{v}) \times \vec{B}=I(\overrightarrow{d l} \times \vec{B})\)

A magnetic monopole (or an isolated magnetic pole) has no existence. There always exists a magnetic dipole with a current loop.

• Magnetic moment of a current loop:

\(\left.\vec{m}=I \vec{A} \text { (SI unit: } \mathrm{A} \mathrm{m}^2\right)\)

• Torque on a magnetic dipole placed in a uniform magnetic field:

\(\vec{\tau}=\vec{m} \times \vec{B}, \text { where } \vec{m}=I N \vec{A}\) and N = number of turns.

• Work done in deflecting a current loop (= a magnetic dipole) in a uniform magnetic field \(\overrightarrow{\mathrm{B}}\):

\(W=m B(1-\cos \theta)\), where 9 is the angle between B and m.

The potential energy of a magnetic dipole in \(\vec{B}\): \(U=-\vec{m} \cdot \vec{B}=-m B \cos \theta\)

• • When \(\theta=0^{\circ}, U_{\min }=-m B\), and the dipole is stable.
  • When \(\theta=180^{\circ}, U_{\max }=m B\), and the dipole is unstable.

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Biot-Savart law:

The magnetic field at P due to a current I is \(\overrightarrow{d B}=\frac{\mu_0}{4 \pi}\left(\frac{I \vec{d} \times \hat{r}}{r^2}\right)\) (expressed in teslas)

or \(d B=\frac{\mu_0}{4 \pi}\left(\frac{I d l \sin \theta}{r^2}\right)\), directed into the paper plane.

• Magnetic field due to a circular current loop:
  • At the centre, \(B=\frac{\mu_0 I}{2 R}\).
  • At a point on its axis, \(\vec{B}=\frac{\mu_0 I R^2 \hat{x}}{2\left(R^2+x^2\right)^{3 / 2}}\)
  • At a distance x » R,

⇒ \(B=\frac{\mu_0 I R^2}{2 x^3}=\frac{2 \mu_0}{4 \pi}\left(\frac{\pi R^2 I}{x^3}\right)=\frac{\mu_0}{4 \pi} \cdot 2\left(\frac{A I}{x^3}\right)=\frac{\mu_0}{4 \pi}\left(\frac{2 m}{x^3}\right)\).

Magnetic Effect of Current NEET Notes for Physics

• • At the centre of an arc, \(B=\frac{\mu_0 I \theta}{4 \pi R}\).

• Magnetic field due to a straight current:
  • Of infinite length: \(B=\frac{\mu_0 I}{2 \pi d}\).
  • Of finite length: \(B=\frac{\mu_0 I}{4 \pi d}\left(\sin \theta_1+\sin \theta_2\right)\).
  • At the centre O, the magnetic field has a zero magnitude and is independent of the angle 0.

• Force on a Current element \(\overrightarrow{I l}\) in magnetic field \(\vec{B}\):

⇒ \(\vec{F}=I \vec{l} \times \vec{B}\).

• Force per unit length between two parallel currents:

⇒ \(F=\frac{\mu_0}{2 \pi}\left(\frac{I_1 I_2}{d}\right)\left(\text { in } \mathrm{N} \mathrm{m}^{-1}\right)\)

• Ampere’s circuital law: \(\oint \vec{B} \cdot \overrightarrow{d l}=\mu_0 I\)
• Magnetic field inside a solenoid: \(B=\frac{\mu_0 N I}{l}=\mu_0 n I\).
• The magnetic field inside a toroid (an endless solenoid):

⇒ \(B=\frac{\mu_0 N I}{2 \pi R}=\mu_0 n I\)

• Velocity selector: A charged particle moving with a velocity \(\vec{v}=v \hat{i}\) and passing through crossed electric and magnetic fields emerges undefeated when

⇒ \(\vec{E}=E(-\hat{j}) \text { and } \vec{B}=B(-\hat{k})\).

The electric force \(\vec{F}_{\text {elec }}=-q(E \hat{j})\) is balanced by the magnetic force \(\vec{F}_{\text {mag }}=q(\vec{v} \times \vec{B})=q v B \hat{i} \times(-\hat{k})=q v B \hat{j}\).

Thus, \(q v B=q E \text { or } v=\frac{E}{B}\).

• Magnetic moment \((\vec{m})\) of a bar magnet:

It is given by \((\vec{m})\) = pole strength x magnetic length = p_m x 2l.

Here \((\vec{m})\) is a vector directed from the SP to the NP (SI unit: A m² ) and pm is positive for the NP and negative for the SP (SI unit: A m).

Electrical And Magnetic Properties NEET

• Magnetic field \((\vec{B})\) due to a bar magnet:
  • At an axial point, \(\vec{B}=\frac{\mu_0}{4 \pi}\left[\frac{2 \vec{m} d}{\left(d^2-l^2\right)^2}\right] \approx \frac{\mu_0}{4 \pi}\left(\frac{2 \vec{m}}{d^3}\right)\).
  • At an equatorial point, \(\vec{B}=\frac{\mu_0}{4 \pi}\left[\frac{-\vec{m}}{\left(d^2+l^2\right)^{3 / 2}}\right] \approx \frac{\mu_0}{4 \pi}\left(\frac{-\vec{m}}{d^3}\right)\).
  • At any point \(\mathrm{P}(r, \theta), B=\frac{\mu_0}{4 \pi}\left(\frac{m}{d^3} \sqrt{1+3 \cos ^2 \theta}\right)\)

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• Torque on a bar magnet in a uniform magnetic field \((\vec{B})\) :

⇒ \(\vec{\tau}=\vec{m} \times \vec{B} \Rightarrow \tau=m B \sin \theta\).

• • Potential energy, \(U=-(\vec{m} \cdot \vec{B})=-m B \cos \theta\)
• The time period of oscillations of a bar magnet in a uniform magnetic field \(\vec{B}\):

⇒ \(T=2 \pi \sqrt{\frac{I}{m B}}\),

where t= moment of inertia about the rotational axis.

Hence, \(B=\frac{4 \pi^2 I}{m}\left(\frac{1}{T}\right)^2=k f^2\) where f = frequency of oscillations.

• Magnetic meridian: It is a vertical plane at a place on the earth’s surface containing a resultant magnetic field at that place.
• Geographical meridian: A vertical plane at a place on the earth’s surface passing through the line joining the geographical north and south is called the geographical meridian of that place.
• Elements of terrestrial magnetism: Three elements are required to completely describe the magnetic field at a place on the earth (both in magnitude and direction). These are listed and explained below.

• • Declination: The angle between the magnetic meridian and the geographical meridian at a place is called fire declination at that place. It is expressed as 0°E or 0W.
  • Dip or inclination (δ): The angle which the earth’s magnetic field \(\vec{B}\) makes with the horizontal line in the magnetic meridian at a place is called the dip (δ) at that place. At the magnetic north and south poles, the dip is 90°, and on the magnetic equator, δ = 0°.
  • Horizontal component (B_H): It is the component of the resultant field B in the horizontal direction in the magnetic meridian at a place.

From the given figure, \(B_{\mathrm{H}}=B \cos \delta \text { and } B_{\mathrm{V}}=B \sin \delta\).

Electrical And Magnetic Properties NEET

∴ \(\tan \delta=\frac{B_{\mathrm{V}}}{B_{\mathrm{H}}} \Rightarrow B=\sqrt{B_{\mathrm{H}}^2+B_{\mathrm{V}}^2} .\).

• Magnetic properties of matter: The magnetism in solids has its origin in the orbital motions and spin rotations of the orbital electrons of the atoms. These motions cause magnetic moments and add up to produce magnetization in solids in the presence of an external magnetic field.
• Intensity of magnetization (I):

⇒ \(I=\frac{\text { magnetic moment }}{\text { volume }}=\frac{\text { (pole strength) }(\text { length })}{(\text { cross-sectional area })(\text { length })}\)

\(=\frac{\text { pole strength }}{\text { cross-sectional area }}\left(\text { SI unit: } \frac{\mathrm{Am}}{\mathrm{m}^2}=\mathrm{Am}^{-1}\right)\)

• Magnetic intensity (H): When a magnetic substance is placed in an external magnetic field B0, it gets magnetized due to the alignments of its atomic dipoles, and the net field inside the material is given by

\(B=B_0+\mu_0 I=\mu_0(H+I), \text { where } B=B_0=\mu_0 H \text {, }\) in free space (I = 0).

Thus, H = \(H=\frac{B}{\mu_0}-I .\). The SI unit of H is the same as that of \(I\left(\mathrm{~A} \mathrm{~m}^{-1}\right)\).

Different expressions for B and H are as follows.

• • At the centre of a circular coil: \(B=\frac{\mu_0 I}{2 R} \Rightarrow H=\frac{B}{\mu_0}=\frac{I}{2 R}.\).
  • Inside a solenoid: \(B=\mu_0 n I \Rightarrow H=n I\).
  • Biot-Savart law: \(\overrightarrow{d B}=\frac{\mu_0}{4 \pi}\left(\frac{I \vec{d} \times \hat{r}}{r^2}\right) \Rightarrow \vec{H}=\frac{\overrightarrow{d B}}{\mu_0}\).
• Magnetic susceptibility \((\chi)\): Magnetic susceptibility indicates the ability of a material to get magnetized when placed in an external magnetizing field. Thus, the intensity of magnetization (l) is proportional to the magnetic intensity (H).

So, \(I \propto H \Rightarrow I=\chi H .\)

Since both I and H are expressed in amperes per metre, \((\chi)\) is dimensionless.

• • For vacuum, I = 0 and \((\chi)\) = 0.
  • For paramagnetic materials, \((\chi)\) is positive.
  • For diamagnetic materials, \((\chi)\) is negative.
• Magnetic permeability, \(\mu=\frac{B}{H}\)

Relation between relative permeability pr and susceptibility \((\chi)\):

The magnetic field inside a material is

⇒ \(B=\mu_0(H+I)=\mu_0(H+\chi H)=\mu_0 H(1+\chi)\).

But \(B=\mu H\).

Electrical And Magnetic Properties NEET

∴ \(\mu=\mu_0(1+\chi)\)

⇒ \(\frac{\mu}{\mu_0}=\mu_{\mathrm{r}}=1+\chi\).

• Curie’s law: With an increase in temperature, the alignment of the elementary dipoles of a magnetic material is reduced, which decreases its magnetization. According to Curie’s law,

⇒ \(\chi \propto \frac{1}{T} \Rightarrow \chi=\frac{C}{T}\) where C is Curie constant.

• Curie temperature (T_c): It is the temperature at which a ferromagnetic material converts into a paramagnetic one. Thus,

⇒ \(\chi=\frac{C^{\prime}}{T-T_C}\) where Tc is Curie point and C is a constant.

• Electromagnetic induction: An electromotive force (emf) is induced whenever a magnetic flux (O) linked with a coil changes with time. This phenomenon is known as electromagnetic induction.
• Magnetic \(\Phi=N A B \cos \theta=N \vec{A} \cdot \vec{B}\) where N = number of turns, A = area within the coil, B = strength of the magnetic field (in tesla), and 0 = angle between the magnetic field and the normal to the area.

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Magnetic flux is expressed in Webers (symbol: Wb)

1 Wb =l T m².

• Faraday’s law: The induced emf in a coil is proportional to the rate of change of the magnetic flux linked with the coil. Thus,

⇒ \(\mathcal{E}=-\frac{d \Phi}{d t}=-N \frac{d \Phi_0}{d t}\),

where N = number of turns and O0 = magnetic flux linked with each turn.

• Motional emf: When a wire moves through a magnetic field so as to cut the field lines, an emf is induced in the wire, and it is called the motional emf.

It has the magnitude ε =\(B l v \cos \theta\), where \(\vec{v}\) = velocity of the wire, l = length of the wire, \(\vec{B}\) = magnetic field and θ = angle between \(\vec{v}\) and \(\hat{n}\) (the unit vector perpendicular to the length of the wire).

Self-inductance (L): Magnetic flux = \(\Phi=L I, \text { induced emf }=\varepsilon=-\frac{d \Phi}{d t}\) = \(-L \frac{d I}{d t}\) and magnetic energy linked with the inductor = \(U=\frac{1}{2} L I^2\) where L is the self-inductance of the inductor. The self-inductance
of a solenoid is \(L=\mu_0 n^2 A l=\frac{\mu_0 N^2 A}{l}\), where N = total number of turns, l = length of the solenoid and A = area of each turn.

The SI unit of self-inductance is the Henry (symbol: H).

• The energy density (u) in a magnetic field \(\vec{B}\):

⇒ \(\left.u=\frac{\text { total energy }(U)}{\text { volume }(V)}=\frac{B^2}{2 \mu_0} \quad \text { (SI unit: } \mathrm{J} \mathrm{m}^{-3}\right)\)

• Mutual inductance (M):

\(\Phi_2=M I_1 \text { and } \varepsilon_2=-M \frac{d I_1}{d t}\), where M is the mutual inductance of two inductors.

The mutual inductance of two solenoids is given by \(M=\frac{\mu_0 N_1 N_2 A}{l}\).

Magnetic Effect of Current formulae for NEET

The SI unit of mutual inductance is the Henry (symbol: H).

• Growth of current in an LR circuit:

⇒ \(I=I_0\left[1-e^{-(R / L) t}\right]\).

• Decay of current in an LR circuit:

⇒ \(I=I_0 e^{-(R / L) t}\).

• Time constant of an LR circuit, \(\tau=\frac{L}{R}\).
• Instantaneous magnetic flux in a coil rotating in a magnetic field:

⇒ \(\Phi=N A B \cos \omega t\)

• Induced emf in a coil:

⇒ \(\mathcal{E}=-\frac{d \Phi}{d t}=N A B \omega \sin \omega t=\varepsilon_0 \sin \omega t\),

where \(\varepsilon_0=\text { peak emf }=N A B \omega\).

• Instantaneous current in an AC circuit:

⇒ \(I=I_0 \sin \left(\omega t+\Phi_0\right)\),

where \(I_0=\text { peak current, } \omega=\text { angular frequency }=2 \pi f \text { and } \Phi_0\) = initial phase.

• Average values of an alternating current (AC):
  • In one complete cycle, \(I_{\mathrm{av}}=\frac{1}{T} \int_0^T I d t=0\).
  • In a half cycle, \(I_{\mathrm{av}}=\frac{1}{T / 2} \int_0^{T / 2} I d t=\frac{2 I_0}{\pi}\).
  • Root-mean-square(or virtual) value,\(I_{\text {rms }}=\frac{I_0}{\sqrt{2}}=\frac{\text { peak value }}{\sqrt{2}}\).
• Reactance (X):
  • Reactance of an inductor: \(X_{\mathrm{L}}=\omega L=2 \pi f L\).
  • Reactance of a capacitor: \(X_C=\frac{1}{\omega C}=\frac{1}{2 \pi f C}\).
• Impedance (Z):
  • Impedance of an LR circuit: \(Z=\sqrt{R^2+\omega^2 L^2}\)
  • Impedance of a CR circuit: \(Z=\sqrt{R^2+\frac{1}{\omega^2 C^2}}\)
  • Impedance of an LCR circuit: \(Z=\sqrt{R^2+\left(\omega L-\frac{1}{\omega C}\right)^2}\)
  • Impedance of an LC circuit: \(Z=\left|X_L-X_C\right|=\left|\omega L-\frac{1}{\omega C}\right|\).
• Phase difference \((\phi)\) between the current and the voltage in an AC circuit:
  • With R only: = 0, i.e., the current (I) and the voltage (V) are in the same phase.
  • With L only: \(\phi=\frac{\pi}{2}\), i.e., the current lags behind the voltage by \(\).
  • With C only: \(\phi=\frac{\pi}{2}\), i.e., the current leads the voltage by \(\frac{\pi}{2}\).
  • With L and R in series: \(\tan \varphi=\frac{\omega L}{R} \text {, i.e., } I \text { lags } V \text { by } \tan ^{-1}\left(\frac{X_L}{R}\right)\)
  • With C and R in series: \(\tan \varphi=\frac{X_C}{R}=\frac{1}{\omega C R}\)i.e., I leads V by \(\tan ^{-1}\left(\frac{\mathrm{X}_{\mathrm{C}}}{R}\right)\)
  • With L and C in series: \(\tan \varphi=\frac{\left|X_{\mathrm{L}}-X_{\mathrm{C}}\right|}{0}=\infty \Rightarrow \varphi=90^{\circ}\),
  • i.e., the current leads by \(\frac{\pi}{2}\) for Xc > XL and the current lags by \(\frac{\pi}{2}\) for XL>Xc.
  • In a series LCR circuit: \(\tan \varphi=\frac{X}{R}=\frac{\left|X_{\mathrm{L}}-X_{\mathrm{C}}\right|}{R}\)
• Power in an AC circuit:
• • True average power = \(\frac{I_0}{\sqrt{2}} \cdot \frac{V_0}{\sqrt{2}} \cos \varphi=(\text { rms power }) \cos \varphi\) (rms power)cos <p.
  • \(\text { Power factor }=\frac{\text { true average power }}{\text { rms power }}=\cos \varphi=\frac{R}{Z}\).
• Electrical resonance: A series RLC circuit is said to be at resonance when the current amplitude \(I_0=\frac{V_0}{Z}\) becomes maximum at a specific frequency called the resonant frequency \(\left(f_{\mathrm{r}}=\frac{1}{2 \pi \sqrt{L C}}\right)\).

“magnetism definition physics “

At resonance,

• • The circuit is purely resistive,
  • Reactance = X = X_L-X_c = 0 ,
  • 1 and 5 are in phase, i.e., \(\varphi=0\),
  • power factor =1 (maximum).
• Q (quality) factor: \(Q=\frac{1}{R} \sqrt{\frac{L}{C}}\)
• LC oscillations: \(\frac{1}{2} L I^2+\frac{Q^2}{2 C}\) = constant, \(\omega=2 \pi f=\frac{1}{\sqrt{L C}} \text { and }\) and Q = Qo cos cof.
• Transformer’s turns ratio, \(\frac{N_{\mathrm{s}}}{N_{\mathrm{p}}}=\frac{V_{\mathrm{s}}}{V_{\mathrm{p}}}=\frac{I_{\mathrm{p}}}{I_{\mathrm{s}}}\).