# Physics 9702 Doubts | Help Page 71

__Question 386: [Electricity > Electric field]__
Two oppositely-charged parallel
metal plates are situated in a vacuum, as shown in Fig.

Plates have length L.

Uniform electric field between
plates has magnitude E. Electric field outside plates is zero.

Positively-charged particle has mass
m and charge +q. Before particle reaches the region between the plates, it is
travelling with speed v parallel to the plates.

Particle passes between plates and
into the region beyond them.

**(a)**

(i) On Fig, draw path of the
particle between the plates and beyond them.

(ii) For the particle in region
between the plates, state expressions, in terms of E, m, q, v and L, as
appropriate, for

1. Force F on the particle

2. time t for particle to cross the
region between the plates

**(b)**

(i) State law of conservation of
linear momentum.

(ii) Use answers in (a)(ii) to state
an expression for change in momentum of the particle.

(iii) Suggest and explain whether
law of conservation of linear momentum applies to the particle moving between
the plates.

**Reference:**

*Past Exam Paper – November 2010 Paper 23 Q7*

__Solution 386:__**(a)**

(i) The path is a reasonable curve
upwards between plates and straight and at a tangent to the curve beyond the
plates

(ii)

1. Force F = Eq

2. {The
force F only causes an acceleration vertically, it does not affect the
horizontal motion.}

Time t = L / v

**(b)**

(i) The law of conservation of
momentum states that the total momentum of a system remains constant

**OR**the total momentum of a system before a collision equals the total momentum after collision
provided no external force acts on
the system.

(ii) Change in momentum, Δp = {Ft =} EqL
/ v

(iii)

EITHER

The charged particle is not an
isolated system, so the law does not apply

OR

The system is {consists of} the particle and the ‘plates’. There is an
equal and opposite Δp {change in momentum} on the plates, so the law
applies.

__Question 387: [Current of Electricity > Power dissipated]__The resistors P, Q and R in the circuit have equal resistance.

The battery, of negligible internal resistance, supplies a total power of 12W.

What is power dissipated by heating in resistor R?

A 2 W B 3 W C 4 W D 6 W

**Reference:**

*Past Exam Paper – June 2010 Paper 11 Q33 & Paper 12 Q34 & Paper 13 Q35*

__Solution 387:__**Answer: A.**

**EITHER**

Power supplied by battery = VI = 12 W
where I is the total current across the circuit and E is the e.m.f. of the
battery.

We need to find the (relative) current
flowing through each resistor and the potential difference across each of them.

Current flows from the positive
terminal of the battery. So, the total current I would flow through resistor P.
At the junction, the current splits equally into 2 since the resistances of Q
and R are equal. So, the currents through Q and R are each I/2.

From Kirchhoff’s law, the
sum of potential difference across any loop is equal to the sum e.m.f of the battery
in the circuit. Also, the p.d. across the components in a parallel combination
is the same.

The equivalent resistance of the
parallel combination of resistors Q and R is half their original resistance.
From the potential divider equation, the p.d. across resistor P is twice that
across the parallel combination (and thus, across Q or R). Let the e.m.f of the
battery be V. The p.d. across P is 2V/3 while the p.d. across Q or R is V/3 (p.d. across P is twice, giving the sum of p.d. in any loop =
V).

Power supplied by battery = VI = 12 W

Resistor R: Current = I/2 p.d. = V/3

Power dissipated in resistor R = (I/2)
(V/3) = (VI) / 6 = 12 / 6 = 2W

**OR**

This could alternatively be done by
calculations.

Let the resistances of resistors P,
Q and R be r each.

Equivalent resistance of parallel
combination = [1/r + 1/r]

^{-1}= 0.5r
Total resistance in circuit, r

_{total}= r + 0.5 = 1.5r = 3r / 2
Total power dissipated in circuit = I

^{2}r_{total}where I is the total current in the circuit.
Total current I = √(Power
/ r

_{total})
As explained before, the total
current I passes through P, but only half of the current passes through Q and
R.

Current through P, I

_{p}= √(Power / r_{total}) = √(12 / 1.5r) = √(8/r)
Half of this current passes through
R.

Current I

_{R}through R = I_{p}/2 = 0.5 √(8/r) = √(2/r)
Power dissipated in resistor R = I

_{R}^{2}r = (2/r) r = 2W

__Question 388: [Gases > Kinetic theory of gases]__**(a)**Kinetic theory of gases is based on some simplifying assumptions.

Molecules of the gas are assumed to
behave as hard elastic identical spheres.

State assumption about ideal gas
molecules based on

(i) nature of their movement

(ii) their volume

**(b)**Cube of volume V contains N molecules of an ideal gas. Each molecule has a component c

_{X}of velocity normal to one side S of the cube, as shown.

Pressure p of the gas due to
component c

_{X}of velocity is given by expression
pV = Nmc

_{X}^{2}
where m is mass of a molecule.

Explain how expression leads to the
relation

pV = (1/3) Nm<c

^{2}>
where <c

^{2}> is mean square speed of the molecules.**(c)**Molecules of an ideal gas have a root-mean-square (r.m.s.) speed of 520 m s

^{–1}at a temperature of 27 °C.

Calculate r.m.s. speed of the
molecules at a temperature of 100 °C.

**Reference:**

*Past Exam Paper – June 2012 Paper 41 Q2*

__Solution 388:__**(a)**

(i)

EITHER The gas molecules are in
random motion

OR The gas molecules have constant
velocity until they hit the wall / other molecules

(ii)

EITHER The (total) volume of the
molecules is negligible compared to the volume of the containing vessel

OR The radius / diameter of a
molecule is negligible compared to the average intermolecular distance.

**(b)**

{Consider a single
molecule of the gas in motion. This molecule will have 3 components of velocity
(since it is able to move in 3 dimensions).}

EITHER A molecule has components of
velocity in 3 directions OR c

^{2}= c_{X}^{2}+ c_{Y}^{2}+ x_{Z}^{2}
{Now, consider all the gas
molecules which are in motion. Since the motion of the gas molecules are random
in nature and the number of molecules is large, the components of velocities of
all the gas molecules can be averaged as <c

_{X}^{2}> = <c_{Y}^{2}> = <c_{Z}^{2}>}
Random motion and averaging, so
<c

_{X}^{2}> = <c_{Y}^{2}> = <c_{Z}^{2}>
{Since <c

_{X}^{2}> = <c_{Y}^{2}> = <c_{Z}^{2}>, each of them can be written as the component of velocities in one specific direction. E.g. <c_{X}^{2}>. Now, replace this in the average of the equation c^{2}= c_{X}^{2}+ c_{Y}^{2}+ x_{Z}^{2}. <c^{2}> = <c_{X}^{2}> + <c_{X}^{2}> + <c_{X}^{2}> = 3<c_{X}^{2}>}
<c

^{2}> = 3<c_{X}^{2}>
{But, since the pressure p
of the gas due to component c

_{X}of velocity is given by expression pV = Nmc_{X}^{2}. <c_{X}^{2}> = <c^{2}> / 3}
So, pV = (1/3) Nm<c

^{2}>**(c)**

{crms = √<c

^{2}>}
<c

^{2}> α T or c_{rms}α √T
(In kelvin,) the temperatures are (27+273
=) 300K and (100+273 =) 373K

{When T = 300K, c

_{rms}= 520ms^{-1}. When T = 373K, c_{rms}= √(373/300) x 520 = 580 ms^{-1}. }
c

_{rms}= 580 ms^{-1}
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