Detailed Notes for JEE preparation on “Magnetic Field for a Straight Conductor and Ampere’s Law”

1. Biot-Savart Law

• Reference: NCERT Physics Class 12, Chapter - 4 (Moving Charges and Magnetism)

• Key points:

  • The magnetic field (\overrightarrow{B}) at a point due to a current-carrying element (Id\overrightarrow{l}) is given by: $$ \overrightarrow{dB} = \frac{\mu_0}{4\pi} \frac{Id\overrightarrow{l} \times\hat{r}}{r^2}$$
  • Where (I) is the current, (d\overrightarrow{l}) is the vector length of the current element, (\overrightarrow{r}) is the position vector from the current element to the observation point, (\hat{r}) is the unit vector along (\overrightarrow{r}), and (\mu_0) is the vacuum permeability ((4\pi \times 10^{-7} \text{T}\cdot \text{m/A})).
  • The direction of (\overrightarrow{B}) is perpendicular to both (d\overrightarrow{l}) and (\overrightarrow{r}), following the right-hand rule.

2. Magnetic Field Lines

• Reference: NCERT Physics Class 12, Chapter - 4 (Moving Charges and Magnetism)

• Key points:

  • Magnetic field lines are imaginary lines used to visualize the direction and strength of the magnetic field.
  • Magnetic field lines always form closed loops and their direction represents the direction of (\overrightarrow{B}) at each point.
  • The closeness of the magnetic field lines indicates the strength of the magnetic field.

3. Magnetic Field Strength

• Reference: NCERT Physics Class 12, Chapter - 4 (Moving Charges and Magnetism)

• Key points:

  • The magnitude of the magnetic field strength (B) at a point due to a straight conductor carrying current (I) is given by: $$ B=\frac{\mu_0 I}{2\pi r}$$
  • Where (r) is the perpendicular distance from the conductor to the observation point.

4. Ampere’s Law

• Reference: NCERT Physics Class 12, Chapter - 4 (Moving Charges and Magnetism)

• Key points:

  • Ampere’s Law states that the magnetic field around a current-carrying wire is proportional to the current and inversely proportional to the distance from the wire.
  • In mathematical form: $$ \oint \overrightarrow{B}\cdot d\overrightarrow{l}=\mu_0 I_{\text{enc}}$$
  • Where (\overrightarrow{B}) is the magnetic field, (\overrightarrow{dl}) is a differential length vector along a closed loop, (I_{enc}) is the net current flowing through the loop in the direction consistent with the right-hand rule, and (\mu_0) is the vacuum permeability.

5. Applications of Ampere’s Law

• Reference: NCERT Physics Class 12, Chapter - 4 (Moving Charges and Magnetism)

• Key points:

  • Ampere’s Law is useful for calculating the magnetic field of various current configurations such as long straight wires, solenoids, and toroids.
  • It can be used to determine the magnetic field both inside and outside the current-carrying structures.

6. Symmetry and Superposition

• Reference: NCERT Physics Class 11, Chapter - 5 (Laws of Motion); Class 12, Chapter - 4 (Moving Charges and Magnetism)

• Key points:

  • Symmetry can simplify magnetic field calculations by identifying regions where the magnetic field is symmetric.
  • The principle of superposition states that the net magnetic field due to multiple current-carrying conductors is the vector sum of the magnetic fields due to each conductor.

7. Magnetic Force between Current-Carrying Conductors

• Reference: NCERT Physics Class 12, Chapter - 4 (Moving Charges and Magnetism)

• Key points:

  • The magnetic force (\overrightarrow{F}) between two parallel current-carrying conductors is given by: $$ \overrightarrow{F}=\frac{\mu_0 I_1I_2}{2\pi d}\overrightarrow{l_1}\times\overrightarrow{l_2}$$
  • Where (I_1) and (I_2) are the currents in the conductors, (d) is the distance between the conductors, and (\overrightarrow{l_1}) and (\overrightarrow{l_2}) are the lengths of the conductors in the direction of the current.
  • The direction of (\overrightarrow{F}) is perpendicular to both (\overrightarrow{l_1}) and (\overrightarrow{l_2}), following the right-hand rule.

Remember to supplement these detailed notes with practice problems and conceptual questions to reinforce your understanding and prepare effectively for JEE exams.