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Fig. 1 The f ield lines of a pointlike charge Qin front of a me tal

plate in compa rison with the field lines o f two cha rges Q

and Q.

ElectricityElectrostaticsEffects of force in an electric field

LD

Physics

Leaflets

Mea suring the force betw een acha rge d s phere a nd a metal pla te

Objects of the experiments

Meas uring the force Fbetw een a c harged sphere and a n earthed metal plate a s a function of the distance dbetween

the centre of the sphere and the metal plate.

Measuring the force F between a charged sphere and an earthed metal plate as a function of the charge Q on the

sphere.

Principles

By electrostatic induction (displacement of charges), a point-

like charge Q at a distance d from an earthed metal plate

generates an exce ss cha rge o f the oppos ite sign on the surfac e

of the me ta l pla te. The c harge Q is, therefore, acted upon byan attractive force towa rds the metal plate.

This a ttrac tive force F is equal to the force that a pointlike

charge Qat a d istance 2dwould exert on the charge Q, that

is,

F=1

40

Q2

(2d)2(I).

This rela tion is illust rate d in Fig . 1: In the c a s e o f an e q uilibrium,

the electric field lines starting from Q meet the metal plate

perpendicularly because a field component parallel to thesurfac e of the plate would lea d to a displac ement of the chargedistribution on the metal plate, which cannot occur in a state

of eq uilibrium. The s a me s hap e o f the field lines is ge nerated

by a mirror cha rge Qplace d a t the mirror ima ge of the pointwhere Q is loca ted w ith respec t to the plate.

In the experiment, the force betw een a cha rged sphere a nd ameta l pla te is mea sured w ith the a im of c onfirming the propor-

tionalities

F1

d2(II)

a nd

FQ2 (III).

In the experiment, ho we ver, the mutual elec tros tatic induction

between the sphere and the metal plate leads to a displace-

ment of charges on the sphere as well which shows up

particularly at small distances d. This d isplac ement ha s thesa me effect that a reduction of the dista nce dwould ha ve, and

it lead s to a n enhancement of the force F.

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Preliminary remarks

Ca rrying o ut this experiment requires pa rticular ca re b ec a use

leakage currents through the insulators may cause chargelosses and thus considerable measuring errors. Moreover,

undes irable effects of electrosta tic induction may influence theresults.

The experiment must be c a rried out in a c los ed , dry room s o

as to prevent charge losses due to high humidity.

Cleaning the insulators of the spheres with distilled water isrecommended because distilled water is the best solvent of

conductive salts on the insulators. In addition, the insulators

should be discharged before every experiment by passing

them s everal times q uickly through a non-bla ckening flame, forexa mple of a buta ne ga s b urner. The pla stic rod a nd the leather

too must be dry and clean.

Setup

The e xperimenta l set up is illustra ted in Fig. 2.

Put the capacitor plate on the insulator from the electro-sta tic a cc es so ries (516 37) onto the vertica lly a djusta blestand, lock with the knurled screw (b), and align the platehorizontally with the levelling screws (d).

Plug a connection lead into the 4-mm hole on the socketof the ca pac itor plate and connect it to the ea rth.

Extend the vertically adjustable stand to maximum heightby turning the height adjustment screw (c), and adjust tozero.

S et the stand rod up in the sta nd ba se, a nd atta ch the forcese nso r (+ F direc tion upwa rd) to the s ta nd rod with the

Leybold multiclamp.

P lug the s phere on a n insulator with a pa ir of plugs (a)intothe socket of the force sensor.

Connect the force sensor to the newtonmeter with themultico re ca ble.

Adjust the height of the force sensor so that the distancebetw een the metal plate and edg e of the sphere is 15 mm(distance dto the ce ntre o f the sphe re: 30 mm).

Carrying out the experiment

Notes:

The measurement is susceptible to impact t hrough interfer-

ences from the vicinity because the forces to be measured are

very small: Avoid vibrations, draught and variations in temp era-

ture.

The new tonmete r must w a rm up at leas t 30 min be fore theexperiment is started: the force sensor being connected,

sw itch the newtonmeter on at the mains s witch on the ba ck ofthe instrument.

a) The force Fas a function of the distance d :

Make the zero compensation by setting the pushbuttonCOMP ENSATION of t he new tonm ete r to S ET.

Cha rge the plas tic rod by rubbing it with the leather.

Cha rge the sphere (a)by touching it with the plas tic rod.

Read the force F and record it together with the distanced.

Increase the distance dwith the height adjustment screw(c)in ste ps o f 2.5-mm up to d= 45 mm. In each ca se, readthe force from the ne wto nmeter and rec ord it tog ether with

the distance.

b) The force Fas a function of the charge Q:

Set the distance dto 35 mm.

Make the zero c ompensa tion of the newtonmeter ag ain.

Rec harge the pla stic rod b y rubb ing it with the lea ther, a ndcha rge the sphere (a)by touching it with the plas tic rod.

Read the force from the new tonmeter and record it as F(Q).

Halve the cha rge b y touching the s phere (a)w ith the s phere

(e), which has the same s ize. Read the force a ga in and record it as F(Q/2).

Ca lculate the ratio o f the two forces .

Disc harge the two s phere a t the capa citor plate, a nd repea tthe experiment s everal times .

Apparatus

1 electrostatic ac ces sories for current ba lanc e 516 37

1 vertica lly ad justab le sta nd . . . . . . . . 516 31

1 newtonmeter . . . . . . . . . . . . . . . 314 251

1 force se nso r . . . . . . . . . . . . . . . . 314 2611 multico re ca ble 6-pole, 1.5 m long . . . . 501 16

1 sta nd rod, 47 cm . . . . . . . . . . . . . 300 421 sta nd ba se, V-shape, 20 cm . . . . . . . 300 02

1 Leybold multiclamp . . . . . . . . . . . . 301 01

1 pla stic rod . . . . . . . . . . . . . . . . . 541 04

1 lea ther . . . . . . . . . . . . . . . . . . . 541 21

connection lea ds

Fig. 2 Experimental setup for the measurement of the force be-

tween a charged s phere and an ea rthed metal plate

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Measuring example

a) Measuring the force Fas a function of the charge Q:

b) Measuring the force Fas a function of the charge Q:

Evaluation

a) The force Fas a function of the distance d :

In Fig . 3, the mea suring va lues of Ta ble 1 a re plotted . The

a ttra ctive force Fbetw een the charged sphere and the ea rthed

metal plate ((turns out to nonlinearly)) when the distance d is

increased.Fig. 4 s how s the me a suring values in the linea rized form

| F|

1

2=f(d). The mea suring va lues lie to go od a pproxima tion

on a straight line, which, however, does not go through theorigin, but intersects the x-axis at d0 = 17 mm. Thus the

proportionality

| F|

1

2dd0o r F1

(dd0)2

is found. This effect c orres pond s to a shift of the distanc e,which goes back to a charge displacement on the sphere

towards the metal plate due to electrostatic induction.

b) Measuring the force Fas a function of the charge Q:Acc ording to Eq. (III), the force s hould b e reduced by a fac tor

of four after the cha rge Qhas been ha lved, that is, one expec ts

F(Q2)

F(Q)=0.25 (IV).

The mea n va lue o f the result in Ta ble 2 is x_

=0.252, and thesta nda rd deviation is = 0.008. Thus (IV) is c onfirmed .

Results

An attractive force arises between a charged sphere and anea rthed meta l pla te d ue to e lec tros ta tic induction. This force

is e q ual to the force exerted b y a mirror cha rge .

Ta ble 1: The force Fas a function of the distance d

d

mm

F

mN30 5.9

32.5 4.3

35 3.1

37.5 2.4

40 1.9

42.5 1.6

45 1.3

Ta ble 2: The force Fmea sured w ith the total charge Q a nd w ith

the halved cha rge (d= 35 mm)

nF(Q)

mN

F(Q/2)

mN

F(Q/2)

F(Q)

1 3.2 0.9 0.28

2 5.0 1.2 0.24

3 3.4 0.8 0.24

4 4.0 1.2 0.30

5 3.0 0.7 0.23

6 3. 6 0.9 0.25

7 4. 6 1.1 0.24

8 3. 7 1.0 0.27

9 3. 9 0.9 0.23

10 4.2 1.0 0.24

30 35 40 45

d

mm

-6

-4

-2

0

FmN

Fig . 3 Th e fo rc e Fbetween a charged sphere and a meta l plate

as a function of the distance d

Fig. 4 Linearization of the curve in Fig. 3 in the form

| F|

1

2=f(d)

LD DIDACTIC G mb H Leyboldstrass e 1 D-50354 Hrth P hone (02233) 604-0 Telefa x (02233) 604-222 E-mail: info@ld-dida ctic.de b y LD DIDAC TIC G mbH P rinted in the Federa l Repub lic o f G erma ny

Technical alterations reserved

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