A 2.0-cm-tall candle flame is 2.0 m from a wall. You take place to havea lens via focal length of 32 cm.

You are watching: How many places can you put the lens to form a well-focused image of the candle flame on the wall?

A. How many kind of places deserve to you put the lens to form a well-concentrated imageof the candle on the wall?

B. For each position, what is the height and orientation of theimage?

### Answer

*A.*

Take the difference between the object and image distance as 2m.

eginalignedv-u &=2 mathrm~m \v &=2 mathrm~m+uendalignedThe elevation of the object is 2.0 cm.

The expression for lens equation is as follows:

frac1f=frac1u+frac1v

The object is put on the left side of the lens. Thus, the above equation is recomposed as follows:

eginaligned frac1f &=frac1-u+frac1v \ &=frac1v-frac1u endaligned

Substitute 2m+u for v.

eginaligned&=fracu(2 m+u) u-frac(2 m+u)(2 m+u) u \&=fracu-2 m-u(2 m+u) u \&=frac-2 m(2 m+u) uendalignedAobtain, substitute 32cm for f.

eginalignedfrac132 mathrm~cmleft(frac10^-2 mathrm~m1 mathrm~cm ight) &=frac-2 mathrm~m(2 mathrm~m+u) u \frac10.32 mathrm~m &=frac-2 mathrm~m(2 mathrm~m+u) u \(2 mathrm~m+u) u &=-2 mathrm~m(0.32 mathrm~m) \u^2+2 u+0.64 &=0endalignedSolve the quadratic equation.

u=-0.4,-1.6 mathrm~m

Hence, the lens have the right to be inserted at 40 cm and also 160 cm to create a well-focused photo of the candle flame on the wall.

**Therefore, The lens have the right to be inserted at two places.**

*B.*

When the distance between the candle and also lens was 160 mathrm~cm. The expression for lens equation is as follows:

frac1f=frac1u+frac1v

Substitute 32 mathrm~cm for f and 160 mathrm~cm for u.

eginaligned frac132 mathrm~cm &=frac1160 mathrm~cm+frac1v \ frac1v &=frac132 mathrm~cm-frac1160 mathrm~cm \ &=frac140 mathrm~cm \ v &=40 mathrm~cm endalignedThe expression for magnification of the lens is as follows:

M=frac-vu

Substitute 40 mathrm~cm for v and also 160 mathrm~cm for u.

eginaligned M &=frac-40 mathrm~cm160 mathrm~cm \ &=-0.25 mathrm~cm endaligned

Anvarious other expression for magnification is as follows:

M=frach_ih_o

Compare the above 2 expressions and also rearvariety the expression for h_i as follows:

frac-vu=frach_mathrmih_mathrmo

h_mathrmi=frac-(v)u h_6

Substitute 40 mathrm~cm for v, 2 mathrm~cm for h_0, and 160 mathrm~cm for u.

h_i=frac-(40 mathrm~cm)160 mathrm~cm(2 mathrm~cm)

=-0.5 mathrm~cm

When the distance between the candle and also lens was 40 mathrm~cm.

The expression for lens equation is as follows:

M=frac-vu

Substitute 160 mathrm~cm for v and 40 mathrm~cm for u.

M=frac-160 mathrm~cm40 mathrm~cm

=-4 mathrm~cm

Another expression for magnification is as follows:

M=frach_ih_o

Compare the above two expressions and also rearselection the expression for h_i as follows:

frac-vu=frach_mathrmih_mathrme

h=frac-(v)u h_ㅇ

Substitute 160 mathrm~cm for $v, 2 mathrm~cm for h_0, and also 40 mathrm~cm for u.

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h_i=frac-(160 mathrm~cm)40 mathrm~cm(2 mathrm~cm)

=-8 mathrm~cm

**Because of this, the height of the picture as soon as the distance between the candle and also lens are 160 mathrm~cm and also 40 mathrm~cm are -0.5 mathrm~cm and -8 mathrm~cm, respectively. In both cases, the magnification is negative, given that both are inverted.**