List of Important Physics derivations for Board Examinations (2024-25)
CBSE: Class-XII

Important Physics derivations for Board Examinations
Chapter 7 – Alternating Current (AC)
1.Using phasor diagram, derive an expression for voltage, current and impedance in LCR series circuit connected with alternating source of emf ɛ=sin(ωt + ф) . Also, deduce power factor of circuit
2.In a series LCR circuit connected to an a.c. source of voltage, ɛ= sinωt. Use phasor diagram to derive an expression for the current in the circuit. Hence, obtain the expression for the power dissipated in the circuit. Show that power dissipated at resonance is maximum.
3.A series LCR circuit is connected to an ac source. Using the phasor diagram, derive the expression for the impedance of the circuit. Plot a graph to show the variation of current with frequency of the source, explaining the nature of its variation.
4.Derive an expression for average power consumed/dissipated in series LCR circuit connected to alternating source in which the phase difference between volage(emf) and current is ф.
5.Define mean/average value of alternating current and show that the average current for half cycle of A.C. is , where Io is peak current value.
6.Define mean/average value of alternating current and show that the average current for full cycle of A.C. is zero , where Io is peak current value.
7.Define r.m.s. value of alternating current and show that the r.m.s. value of current for half cycle of A.C. is , where Io is peak current value.
8.Define r.m.s. value of alternating current and show that the r.m.s. value of current for full cycle of A.C. is , where Io is peak current value.
9.Show that average power dissipated in pure inductor is zero when it is connected to A.C. supply . [2 Marks]
10.Show that average power dissipated in pure capacitor is zero when it is connected to A.C. supply . [2 Marks]
9.Show that average power dissipated in pure inductor is zero when it is connected to A.C. supply . [3 Marks]
10.Show that average power dissipated in pure capacitor is zero when it is connected to A.C. supply . [3 Marks]
11.Show that current leads voltage (emf) by phase angle in pure capacitive circuit with capacitance C when it is connected to A.C. source. [3 Marks]
12.Show that the voltage (emf) leads current by phase angle in pure inductive circuit with capacitance C when it is connected to A.C. source. [3 Marks]
Important Physics derivations for Board Examinations
Chapter 6 – Electromagnetic Induction (EMI)6
1.A conducting rod of length ℓ is kept perpendicular to uniform magnetic field \overrightarrow B. It is moved along the magnetic field with a velocity \overrightarrow v. Derive the expression of e.m.f. (motional e.m.f.) induced in the conductor.
2.A metallic rod MN of length ℓ is rotated with angular velocity \omega about an axis passing through one of its end and perpendicular to the plane of the paper, in uniform magnetic field \overrightarrow B as shown in figure. Derive an expression for the induced emf (motional e.m.f.) developed between the end points M and N.
3. The figure shows a rectangular conducting frame MNOP of resistance R placed partly in a perpendicular magnetic field \overrightarrow B and moved with velocity \overrightarrow v as shown in the figure.
Obtain the expressions for the
(a) induced current in the loop
(b) force acting on the arm ‘ON’ and its direction, and
(c) power required to move the frame to get a steady emf induced between the arms MN and PO.
4.Two concentic circular coils X and Y of radii r_1 and r_2 (r_1>>r_2) having N_1 and N_2 turns respectively are placed coaxially with centres coinciding. Obtain an expression for
(i) the mutual inductance for the arrangement, and
(ii) the magnetic flux linked with coil Y when current I flows through coil X.
5.Obtain the expression for the mutual inductance of two long co-axial solenoids S_1 and S_2 wound one over the other , each of length L and radii r_1 and r_2 and n_1 and n_2 number of turns per unit length , when a current I is set up in the outer solenoid S_2
6. Define self-inductance of a coil. Derive the expression for magnetic energy stored in an inductor L connected across a source of emf to build up a current I through it.
7.Define self-inductance of a coil. Derive the expression for self-inductance of a solenoid of length L and and r, having N turns.
8. A rectangular coil of area A, having number of turns N is rotated at ‘f ‘ revolutions per second in a uniform magnetic field B, the field being perpendicular to the coil. Prove that the maximum emf induced in the coil is 2\pi fBAN
9.A metallic rod MN of length ℓ moves with linear velocity \overrightarrow v , perpendicular to uniform magnetic field \overrightarrow B as shown in figure. Derive an expression for the induced emf (motional e.m.f.) developed between the end points M and N.
Important Physics derivations for Board Examinations
Chapter 5 – Magnetism and Magnetic Materials
1.Derive relationship (\mu_{r\;}=\;1+\;\chi_m) between magnetic susceptibility \chi_m and relative permeability \mu_r.
2. Show that a current carrying solenoid is as equivalent to a tiny bar magnet.
Important Physics derivations for Board Examinations
Chapter 4 – Moving Charges and Magnetism
1.(i) State Biot – Savart law in vector form expressing the magnetic field\overrightarrow B due to an element \overrightarrow {dl} carrying current I at a distance \overrightarrow r from the element.
(ii) Write the expression for the magnitude of the magnetic field at the centre of a circular loop of radius r carrying a steady current I. Draw the field lines due to the current loop.
2. Use Biot-Savart law to derive the expression for magnetic field B at a point P on the axis (distance x from centre) of a circular coil of radius ‘r’ carrying current ‘I’ and hence find the magnetic field at the centre ‘O’ of the circular coil carrying current.
3. Using Ampere’s circuital law, obtain an expression for the magnetic field due a infinitely long straight wire carrying current ‘I’.
4.Using Ampere’s circuital law, obtain an expression for the magnetic field along the axis of a current carrying solenoid of length l and having N number of turns.
5.Derive the expression for force per unit length between two long straight parallel current carrying conductors. Hence define one ampere.
6.Two identical circular loops, P and Q, each of radius r and carrying current I and 2I respectively are lying in parallel planes such that they have a common axis. The direction of current in both the loops is clockwise as seen from O which is equidistant from both the loops. Obtain the expression for the magnitude of the net magnetic field at point O.
7.Two identical circular loops, P and Q, each of radius r and carrying equal currents are kept in the parallel planes having a common axis passing through O. The direction of current in P is clockwise and in Q is anti-clockwise as seen from O which is equidistant from the loops P and Q. Obtain the expression for the magnitude of the net magnetic field at O.
8. A rectangular coil PQRS of sides ‘l’ and ‘b’ carrying a current I is subjected to a uniform magnetic field \overrightarrow B acting perpendicular to its plane. Obtain the expression for the torque acting on it.
9.Deduce the expression for the magnetic dipole moment of an electron orbiting around the central nucleus.
10.Describe the working principle of a moving coil galvanometer. Why is it necessary to use
(i) a radial magnetic field and
(ii) a cylindrical soft iron core in a galvanometer? Write the expression for current sensitivity of the galvanometer.
Can a galvanometer as such be used for measuring the current?
11.(a) Discuss the conversion of galvanometer to ammeter which can measure current ranging from 0 to I. Deduce the expression for ammeter current I if galvanometer can allow maximum current I_g to pass through itself.
(b) Explain, giving reasons, the basic difference in converting a galvanometer into
(i) a voltmeter and
(ii) an ammeter.
12.(a) Discuss the conversion of galvanometer to voltmeter which can measure voltage ranging from 0 to V. Deduce the expression for potential difference V that it can measure if galvanometer can allow maximum current I_g to pass through itself.
(b) Explain, giving reasons, the basic difference in converting a galvanometer into
(i) a voltmeter and
(ii) an ammeter.
13.(a) Use Biot-Savart law to derive the expression for the magnetic field due to a circular coil of radius R having N turns at a point on the axis at a distance ‘x’ from its centre. Draw the magnetic field lines due to this coil.
(b) A current ‘I’ enters a uniform circular loop of radius ‘R’ at point M and flows out at N as shown in the figure.
Obtain the net magnetic field at the centre of the loop.
Important Physics derivations for Board Examinations
Chapter 3 – Current Electricity
1.Derive an expression for drift velocity of free electrons in a conductor in terms of relaxation time.
2.Explain the term ‘drift velocity’ of electrons in a conductor. Hence obtain the expression for the current through a conductor in terms of ‘drift velocity’
3.Derive an expression for the resistivity of a good conductor, in terms of the relaxation time of electrons.
4.(i) Define the term drift velocity.
(ii) On the basis of electron drift, derive an expression for resistivity of a conductor in terms of number density of free electrons and relaxation time. On what factors does resistivity of a conductor depend?
(OR)
Derive an expression for the resistivity \rho=\frac m{ne^2\tau} a good conductor, in terms of the relaxation time of electrons.
5.Use Kirchhoff’s rules to derive conditions for the balanced Wheatstone bridge.
6.Using the concept of drift velocity of charge carriers in a conductor, deduce the relationship between current density and resistivity/conductivity of the conductor.
7.Derive an expression for the current density of a conductor in terms of the drift speed of electrons.
8.A number of identical cells n, each of emf e, internal resistance r connected in series are charged by a d.c. source of emf elr using a resistor R.
(i) Draw the circuit arrangement.
(ii) Deduce the expressions for
(a) the charging current and (b) the potential difference across the combination of the cells.
9.Find the relation between drift velocity and relaxation time of charge carriers in a conductor. A conductor of length L is connected to a d,c. source of emf ‘E’. If the length of the conductor is tripled by stretching it, keeping ‘E’ constant, explain how its drift velocity would be affected.