Pls someone ans it
Prove mirror formual when object is placed at centre of curvature of concave mirror.
Answers
Mirror Formula for Convex Mirror when Real Image is formed
Let AB be an object lying beyond the focus of a concave mirror. A ray of light BL after reflecting from the concave mirror passes through the principal axis at F and goes along LB’. Another ray from B passes through the centre of curvature © and incident normally on the mirror at point M. after reflection, this ray retraces its path and meets LB’ at B’. So A’B’ is the real image of the object AB. Draw LN perpendicular on the principle axis.
NowΔ‘sNLF and A’B’F are similar, thereforeA′B′NLSince aperture of the concave mirror is small, so point N lies very close to P.NFAlsoNLequation(i)becomes,A′B′ABAlsoΔ‘sABC and A’B’C are similar, therefore,A′B′ABFrom equation(ii)and(iii),we getA′FPFSince all the distances are measured from the pole of the mirror, so=A′FNF…(i)=PF=AB=A′FPF…(ii)=A′CAC…(iii)=A′CAC…(iv)
A′F=PA′−PFA′C=PC−PA′AC=PA−PC⎫⎭⎬…(i)
Substituting the values of equation(v)in equation,(iv)we getPA′–PFPFApplying sign convention,PA′=v,PF=f,PC=R=2f(∴R=2f)PAHence equation(vi)becomesv−ffuv–2fv–uf+2f2uvDividing by uvf, we getuvuvf1f=PC–PA′PA−PC…(vi)=u=2f–vu–2f=2f2–vt=uf+vf=ufuvf+vfuvf=1u+1v
f
Mirror Formula for Concave Mirror when Virtual Image is formed
When an object is placed between the pole and the focus of a concave mirror, erect and enlarges image formed behind the mirror as shown in figure. Draw LN perpendicular on the principle axis.
NowΔ‘sNLF and A’B’F are similar, thereforeA′B′NLSince aperture of the concave mirror is small, so point N lies very close to P.∴NF=PFandNL=ABAlsoΔ‘sABC and A’B’C are similar, therefore,A′B′AB=A′FNF…(i)=A′CAC=PA′+PCPC–PA…(iii)
From equation(ii)and(iii),we getPA′+PFPFFrom equation(ii)and(iii),we getApplying sign conventionPA′=−v,PF=f,PC=R=2f(∵R=2f,PA=u)∴Equation(iv)becomes−v+ffor,−2vf+uv+2f2–ufor,uv=PA′+PCPC−PA…(iv)=–v−2f2f−u=−vf+2f2=uf+vf
Dividing by uvf, we getuvuvf1f=ufuvf+vfuvf=1u+1v
f
Mirror Formula for Convex Mirror
Let AB be an object lying on the principle axis of the convex mirror of small aperture. A’B’ is the virtual image of the object lying behind the convex mirror as shown in the figure.
Draw LN perpendicular on the principal axis.
NowΔ‘sNLF and A’B’F are similar, thereforeA′B′NLSince aperture of the concave mirror is small, so point N lies very close to P.∴NF=PFandLN=ABAlsoΔ‘sABC and A’B’C are similar, therefore,A′B′AB=A′FNF…(i)=A′CAC=PC−PA′PA+PC…(iii)
From equation(ii)and(iii),we getPF−PA′PFApplying sign conventionPA′=−v,PF=−f,PC=−R=−2f,PA=u(∵R=2f,)∴Equation(iv)becomes−f+vfor,−uf+2f2+uv–2vf=2f2–vfor,uv=PC−PA′PA+PC…(iv)=−2f+vu−2f=uf+vf
Dividing by uvf, we getuvuvf1f=ufuvf+vfuvf=1u+1v
Relation between R and f Parabolic Mirror and Magnification