Magnetic Fields: Physics 9702 (Cambridge International AS & A Level)

A straight horizontal wire $XY$ carries a current of $3.2\text{ A}$ from $X$ to $Y$ (from left to right, as viewed on this page) and has a length of $0.25\text{ m}$ within the field region. The wire is held at right angles to a uniform magnetic field of flux density $0.45\text{ T}$, directed into the plane of the page.

Using Fleming's left-hand rule, what is the direction of the magnetic force on the wire?

A straight wire of length $0.40\text{ m}$ carries a current of $5.0\text{ A}$ and is held at right angles to a uniform magnetic field. The wire experiences a magnetic force of $0.60\text{ N}$.

(a) Calculate the magnetic flux density $B$ of the field. [2]

(b) The current in the wire is now doubled to $10.0\text{ A}$, with the wire still perpendicular to the same field. State and explain the effect this has on the magnetic force on the wire, giving its new value. [2]

(c) The wire, now carrying its original current of $5.0\text{ A}$ again, is turned so that it makes an angle of $30°$ with the magnetic field instead of being perpendicular to it. Calculate the new magnetic force on the wire. [2]

An electron (mass $m=9.11\times10^{-31}\text{ kg}$, charge of magnitude $e=1.60\times10^{-19}\text{ C}$) travels through a vacuum at a constant speed of $v=2.0\times10^{6}\text{ m s}^{-1}$, moving horizontally to the right across the page. It enters a region containing a uniform magnetic field of flux density $B=8.0\times10^{-4}\text{ T}$, directed into the plane of the page, at right angles to its velocity.

(a) Calculate the magnitude of the magnetic force acting on the electron as it enters the field. [2]

(b) State the initial direction of this force on the electron, explaining your reasoning with reference to the electron's negative charge. Explain why this force causes the electron to travel in a circular path at constant speed, and calculate the radius of this path. [4]

(c) Calculate the time taken for the electron to complete one full revolution of its circular path. [2]

A velocity selector uses a uniform electric field of strength $2.4\times10^{4}\text{ V m}^{-1}$ together with a uniform magnetic field of flux density $6.0\times10^{-3}\text{ T}$. The two fields are perpendicular to each other, and both are perpendicular to the initial velocity of a charged particle entering the selector, arranged so that the electric force and the magnetic force on the particle act in opposite directions.

Only particles travelling at one particular speed pass straight through the selector without being deflected. What is this speed?

A flat, rectangular search coil has $N=250$ turns and a cross-sectional area of $A=4.0\times10^{-4}\text{ m}^2$. The coil is initially held with its plane perpendicular to a uniform magnetic field of flux density $B_1=0.080\text{ T}$, so that the field passes straight through the coil.

(a) State what is meant by magnetic flux, and calculate the magnetic flux $\Phi$ through the coil in this initial position. [2]

(b) Calculate the flux linkage of the coil in this initial position. [1]

(c) The coil is then pulled out of the field over a time interval of $0.050\text{ s}$, during which the flux density through the coil falls uniformly from $0.080\text{ T}$ to zero. Use Faraday's law to calculate the average e.m.f. induced in the coil during this time. [3]

(d) State Lenz's law, and use it to describe the direction of the induced current in the coil, relative to the original magnetic field, while the coil is being removed. [2]

A proton (charge $+e=1.60\times10^{-19}\text{ C}$) travels at a constant speed of $v=1.0\times10^{6}\text{ m s}^{-1}$ vertically downward through a region of uniform magnetic field. The field has a flux density of $B=0.60\text{ T}$ and is directed out of the page, toward the reader.

What is the direction of the magnetic force on the proton at this instant?

A beam of protons (mass $m_p=1.67\times10^{-27}\text{ kg}$, charge $+e=1.60\times10^{-19}\text{ C}$) enters a velocity selector. Inside the selector, a uniform electric field of strength $E=3.6\times10^{5}\text{ V m}^{-1}$ and a uniform magnetic field of flux density $B_1=0.45\text{ T}$ are perpendicular to each other, and both are perpendicular to the initial velocity of the protons, arranged so that the electric force and the magnetic force on a proton act in opposite directions.

(a) Calculate the speed $v$ of the protons that pass straight through the selector without being deflected. [2]

(b) These protons then leave the selector and enter a separate region containing only a uniform magnetic field of flux density $B_2=0.20\text{ T}$, directed at right angles to their velocity. Calculate the radius of the circular path followed by the protons in this second region. [3]

(c) Calculate the time taken for a proton to complete a quarter of a full revolution of this circular path. [3]

A rectangular coil is made from $N=60$ turns of wire, wound so that each turn has one side of length $L=0.12\text{ m}$ lying within a uniform magnetic field of flux density $B=0.25\text{ T}$. This side of the coil carries a current of $I=1.5\text{ A}$ and is initially held at right angles to the field.

(a) Calculate the magnetic force acting on this side of a single turn of the coil. [2]

(b) Hence calculate the total magnetic force on this side of the coil, taking into account all $N=60$ turns. [2]

(c) The coil is now rotated so that this side makes an angle of $40°$ with the magnetic field, instead of being at right angles to it, while the current stays at $1.5\text{ A}$. Calculate the new total magnetic force on this side of the coil. [3]

A bar magnet is held above a horizontal circular coil of wire, with its north pole facing downward toward the coil. The coil is connected to a sensitive ammeter. The magnet is released and falls, north pole first, straight down toward the coil.

While the magnet is still above the coil and approaching it (before it reaches the coil), what is the direction of the current induced in the coil, when viewed from above?

A straight conducting rod $PQ$ of length $L=0.30\text{ m}$ lies across two long, parallel, horizontal conducting rails separated by this same distance. The rails are connected together at one end by a fixed resistor of resistance $R=5.0\ \Omega$. The whole arrangement lies in a uniform magnetic field of flux density $B=0.40\text{ T}$, directed vertically, perpendicular to the plane of the rails. The rod is pulled along the rails at a constant speed of $v=2.5\text{ m s}^{-1}$, perpendicular to its own length, so that the area enclosed by the rails, the rod and the resistor increases uniformly with time.

(a) Show that the e.m.f. induced in the circuit is given by $\varepsilon=BLv$, and calculate its value for this rod. [3]

(b) Calculate the current in the circuit while the rod moves at this constant speed. [2]

(c) State Lenz's law, and use it to explain the direction of the magnetic force that this induced current exerts on the rod, and hence explain why an external force must be applied to the rod to keep it moving at constant speed. [3]

(d) Calculate the magnitude of the external force that must be applied to the rod to keep it moving at this constant speed. [2]