Magnetic Field Due to a Current Element, Biot-Savart Law

A magnetic field set up using Helmholtz coils is uniform in a small region and has a magnitude of $0.75 \mathrm{T}$. In the same region, a uniform electrostatic field is maintained in a direction normal to the common axis of the coils. A narrow beam of (single species) charged particles all accelerated through $15 \mathrm{kV}$ enters this region in a direction perpendicular to both the axis of the coils and the electrostatic field. If the beam remains undeflected when the electrostatic field is $9\times {10}^{-5} \mathrm{V}{\mathrm{m}}^{-1}$, make a simple guess as to what the beam contains. Why is the answer not unique?

An electron emitted by a heated cathode and accelerating through a potential difference of $2.0 \mathrm{kV}$, enters a region of a uniform magnetic field of $0.15 \mathrm{T}$, determine the trajectory of the electron if it makes an angle of $30°$ with the initial velocity.

An electron emitted by a heated cathode and accelerated through a potential difference of $2.0 \mathrm{kV}$ enters a region with the uniform magnetic field of $0.15 \mathrm{T}$. Determine the trajectory of the electron if the field is transverse to its initial velocity.

An electron is moving west to east enters a chamber having a uniform electrostatic field in the north to south direction. Specify the direction in which a uniform magnetic field should be set up to prevent the electron from deflecting from its straight-line path.

A charged particle enters an environment of a strong and non-uniform magnetic field varying from point to point both in magnitude and direction and comes out of it following a complicated trajectory. Would its final speed equal the initial speed if it suffered no collisions with the environment?

A magnetic field that varies in magnitude from point to point but has a constant direction (east to west) is set up in a chamber. A charged particle enters the chamber and travels undeflected along a straight path with constant speed. What can you say about the initial velocity of the particle?

A toroid has a core (non-ferromagnetic) of inner radius $25 \mathrm{cm}$ and outer radius $26 \mathrm{cm}$, around which $3500$ turns of a wire are wound. If the current in the wire is $11 \mathrm{A}$, what is the magnetic field outside the toroid, inside the core of the toroid? Take, ${\mu }_0 = 4\pi \times {10}^{-7} \mathrm{H}/\mathrm{m}$

Consider two parallel coaxial circular coils of equal radius $R$, and number of turns $N$, carrying equal currents in the same direction, and separated by a distance $R$. Show that the field on the axis around the midpoint between the coils is uniform over a distance that is small as compared to $R$, and is given by,
$B=0.72\frac{{\mathrm{\mu }}_{0}NI}{R}$

For a circular coil of radius R and N turns carrying current I; the magnitude of the magnetic field at a point on its axis at a distance $x$ from its centre is given by,
$B=\frac{{\mathrm{\mu }}_{0}I{R}^{2}N}{2{(X^2+R^2)}^{\frac{3}{2}}}$
Show that this reduces to the familiar result for the field at the centre of the coil.

A magnetic field of $100 \mathrm{G}$($1\mathrm{}\mathrm{G}=\mathrm{}{10}^{-4}\mathrm{}\mathrm{T}$) is required, which is uniform in a region of linear dimension about $10 \mathrm{cm}$ and area of cross-section about ${10}^{-3} {\mathrm{m}}^2$ . The maximum current-carrying capacity of a given coil of wire is $15 \mathrm{A}$, and the number of turns per unit length that can be wound around a core is at most $1000 \mathrm{turns} {\mathrm{m}}^{-1}$. Suggest some appropriate design particulars of a solenoid for the required purpose. Assume the core is not ferromagnetic.

Two concentric circular coils X and Y of radii $16 \mathrm{cm}$ and $10 \mathrm{cm}$, respectively, lie in the same vertical plane containing the north to south direction. Coil X has $20$ turns and carries a current of $16 \mathrm{A}$; coil Y has $25 \mathrm{turns}$ and carries a current of $18 \mathrm{A}$. The sense of the current in X is anticlockwise, and clockwise in Y, for an observer looking at the coils facing west. Give the magnitude and direction of the net magnetic field due to the coils at their centre.

A circular coil of $30$ turns and radius $8.0 \mathrm{cm}$ carrying a current of $6.0 \mathrm{A}$ is suspended vertically in a uniform horizontal magnetic field of magnitude $1.0 \mathrm{T}$. The field lines make an angle of $60°$ with the normal of the coil. A counter torque is applied to prevent the coil from turning. Would your answer change, if the circular coil were replaced by a planar coil of some irregular shape that encloses the same area? (All other particulars are also unaltered.)

 A circular coil of $30$ turns and radius $8.0 \mathrm{cm}$ carrying a current of $6.0 \mathrm{A}$ is suspended vertically in a uniform horizontal magnetic field of magnitude $1.0 \mathrm{T}$. The field lines make an angle of $60°$with the normal of the coil. Calculate the magnitude of the counter torque that must be applied to prevent the coil from turning.

Obtain the frequency of revolution of the electron, in a magnetic field, in its circular orbit. Does the answer depend on the speed of the electron? Explain.

In a chamber, a uniform magnetic field of $6.5 \mathrm{G}$ ( $1\mathrm{G}={10}^{-4} \mathrm{T}$ ) is maintained. An electron is shot into the field with a speed of $4.8\times {10}^6 \mathrm{m}{\mathrm{s}}^{-1}$ normal to the field. Explain why the path of the electron is a circle. Determine the radius of the circular orbit.
( $e=1.6\times {10}^{-19} \mathrm{C}, m_e=9.1\times {10}^{-31} \mathrm{kg}$)

Two moving coil meters, $M_{\mathrm{1}}$ and $M_{\mathrm{2}}$ have the following particulars: $R_{\mathrm{1}}=10 \mathrm{\Omega }$, $N_{\mathrm{1}}=30$, $A_{\mathrm{1}}=3.6\times {10}^{-3} {\mathrm{m}}^2$, $B_{\mathrm{1}}=0.25\hspace{0.17em}\mathrm{T}$, $R_{\mathrm{2}}=14 \mathrm{\Omega }$, $N_{\mathrm{2}}=42$, $A_{\mathrm{2}}=1.8\times {10}^{-3} {\mathrm{m}}^2$, $B_{\mathrm{2}}=0.50\hspace{0.17em}\mathrm{T}$.
(The spring constants are identical for the two meters). Determine the ratio of current sensitivity and voltage sensitivity of $M_{\mathrm{2}}$ and $M_{\mathrm{1}}$.

A square coil of side $10 \mathrm{cm}$ consists of $20 \mathrm{turns}$ and carries a current of $12 \mathrm{A}$. The coil is suspended vertically, and the normal to the plane of the coil makes an angle of $30°$ with the direction of a uniform horizontal magnetic field of magnitude $0.80 \mathrm{T}$. What is the magnitude of torque experienced by the coil.

A closely wound solenoid $80 \mathrm{cm}$ long has $5$ layers of windings of $400$ turns each. The diameter of the solenoid is $1.8 \mathrm{cm}$. If the current carried is $8.0 \mathrm{A}$, estimate the magnitude of $B$ inside the solenoid near its centre.

Two long and parallel straight wires A and B carrying currents of $8.0 \mathrm{A}$ and $5.0 \mathrm{A}$ in the same direction are separated by a distance of $4.0 \mathrm{cm}$. Estimate the force on a $10 \mathrm{cm}$ section of wire A.

A $3.0 \mathrm{cm}$ wire carrying a current of $10 \mathrm{A}$is placed inside a solenoid perpendicular to its axis. The magnetic field inside the solenoid is given to be $0·27 \mathrm{T}$. What is the magnetic force on the wire?