Magnetism flash cards

1. 1.State the Biot-Savart law for the magnetic field due to a current element.

   $dB = \dfrac{\mu_0}{4\pi}\dfrac{I\,dl\sin\theta}{r^2}$, where $\theta$ is the angle between the current element $I\,dl$ and the position vector $r$. In vector form $d\vec{B} = \dfrac{\mu_0}{4\pi}\dfrac{I\,d\vec{l}\times\hat{r}}{r^2}$.

   Hint: Field of a tiny current element; inverse square, sine of angle.

2. 2.What is the value and SI unit of $\mu_0$, the permeability of free space?

   $\mu_0 = 4\pi\times10^{-7}\ \text{T m A}^{-1}$ (also written $\text{Wb A}^{-1}\text{m}^{-1}$).

   Hint: Contains $4\pi\times10^{-7}$.

3. 3.In the Biot-Savart law, in which direction does $d\vec{B}$ point?

   Along $d\vec{l}\times\hat{r}$, i.e. perpendicular to the plane containing the current element $d\vec{l}$ and the position vector $\hat{r}$, given by the right-hand rule.

   Hint: Cross product direction, perpendicular to both.

4. 4.For a point on the axis of a current element, what is $dB$?

   Zero, because $\theta = 0$ (or $180^\circ$) so $\sin\theta = 0$. A current element produces no field along its own direction.

   Hint: $\sin 0 = 0$.

5. 5.Compare Biot-Savart law with Coulomb's law — one key similarity and one difference.

   Similarity: both are inverse-square laws ($1/r^2$). Difference: Coulomb's field is along the line joining charges (central), while Biot-Savart field is perpendicular to both $d\vec{l}$ and $\hat{r}$ (not central); also magnetism needs a current (moving charge).

   Hint: Both $1/r^2$; one central, one perpendicular.

6. 6.Magnetic field at the centre of a circular loop of radius $R$ carrying current $I$?

   $B = \dfrac{\mu_0 I}{2R}$, directed along the axis (perpendicular to the plane of the loop).

   Hint: $\mu_0 I$ over $2R$.

7. 7.Field at the centre of a coil of $N$ turns, radius $R$, current $I$?

   $B = \dfrac{\mu_0 N I}{2R}$ — the single-loop result multiplied by the number of turns $N$.

   Hint: Multiply single-loop by $N$.

8. 8.Magnetic field on the axis of a circular loop at distance $x$ from centre?

   $B = \dfrac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}$ (for $N$ turns, multiply by $N$).

   Hint: Denominator has power $3/2$.

9. 9.For a circular loop, what happens to the axial field far away ($x\gg R$)?

   $B \approx \dfrac{\mu_0 I R^2}{2x^3} = \dfrac{\mu_0}{4\pi}\dfrac{2m}{x^3}$ with $m = I\pi R^2$ — it falls as $1/x^3$, like a magnetic dipole.

   Hint: $1/x^3$ dipole field.

10. 10.Field at the centre due to a circular arc subtending angle $\phi$ (radians) at the centre?

    $B = \dfrac{\mu_0 I}{4\pi R}\phi$. For a full circle $\phi = 2\pi$, giving $\dfrac{\mu_0 I}{2R}$; for a semicircle $\dfrac{\mu_0 I}{4R}$.

    Hint: Fraction $\phi/2\pi$ of full loop.

11. 11.Magnetic field due to a long straight wire at perpendicular distance $r$?

    $B = \dfrac{\mu_0 I}{2\pi r}$, with circular field lines around the wire; direction by right-hand thumb rule.

    Hint: $\mu_0 I$ over $2\pi r$.

12. 12.State the right-hand thumb rule for a straight current-carrying wire.

    If the thumb of the right hand points along the current, the curled fingers give the direction of the circular magnetic field lines around the wire.

    Hint: Thumb = current, fingers = field.

13. 13.How does $B$ vary with distance for (a) a long straight wire and (b) far from a loop/dipole?

    (a) Straight wire: $B \propto 1/r$. (b) Far from a dipole/loop on axis: $B \propto 1/r^3$.

    Hint: $1/r$ vs $1/r^3$.

14. 14.State Ampere's circuital law.

    $\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{enc}$ — the line integral of $\vec{B}$ around a closed loop equals $\mu_0$ times the net current enclosed by the loop.

    Hint: Closed loop integral = $\mu_0 I_{enc}$.

15. 15.Use Ampere's law to find the field of a long straight wire.

    Take a circular loop of radius $r$: $\oint B\,dl = B(2\pi r) = \mu_0 I$, so $B = \dfrac{\mu_0 I}{2\pi r}$.

    Hint: $B\cdot 2\pi r = \mu_0 I$.

16. 16.Magnetic field inside a long solenoid (n turns per unit length)?

    $B = \mu_0 n I$, uniform and directed along the axis; independent of the solenoid's radius.

    Hint: $\mu_0 n I$; uniform inside.

17. 17.What is the magnetic field outside an ideal long solenoid?

    Approximately zero — the field is essentially confined inside; an ideal long solenoid behaves like a bar magnet.

    Hint: Nearly zero outside.

18. 18.Magnetic field inside a toroid with $N$ turns, mean radius $r$?

    $B = \dfrac{\mu_0 N I}{2\pi r} = \mu_0 n I$ where $n = N/(2\pi r)$. Field is confined within the toroid; zero outside and in the central cavity.

    Hint: Like a solenoid bent into a ring.

19. 19.How is $n$ (turns per unit length) related to total turns $N$ in a solenoid of length $L$?

    $n = N/L$. So $B = \mu_0 n I = \mu_0 N I / L$.

    Hint: $n = N/L$.

20. 20.Field at the end of a long solenoid (on axis, at the mouth)?

    $B = \dfrac{\mu_0 n I}{2}$ — exactly half the field at the centre of the solenoid.

    Hint: Half the central value.

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