7.Two polarizers are oriented at 42.0^circ to one another .Light polarized at : 21.0^circ angle to each polarizer passes through both.What is the transmitted inten. sity (0,0)[1] ?

To find the transmitted intensity, we can use Malus's law, which relates the intensity of light passing through a polarizer to the angle between the light's initial polarization direction and the axis of the polarizer.<br /><br />Malus's law states that the intensity of light passing through a polarizer is given by:<br /><br />\[ I = I_0 \cdot \cos^2(\theta) \]<br /><br />where:<br />- \( I \) is the transmitted intensity,<br />- \( I_0 \) is the initial intensity of the light,<br />- \( \theta \) is the angle between the light's initial polarization direction and the axis of the polarizer.<br /><br />In this case, the light is polarized at an angle of \( 21.0^{\circ} \) to each polarizer, and the two polarizers are oriented at an angle of \( 42.0^{\circ} \) to one another. To find the transmitted intensity, we need to consider the effect of both polarizers.<br /><br />Let's denote the initial intensity of the light as \( I_0 \). After passing through the first polarizer, the intensity of the light becomes:<br /><br />\[ I_1 = I_0 \cdot \cos^2(21.0^{\circ}) \]<br /><br />Now, the light passes through the second polarizer, which is oriented at an angle of \( 42.0^{\circ} \) to the first polarizer. The angle between the light's polarization direction and the axis of the second polarizer is \( 21.0^{\circ} + 42.0^{\circ} = 63.0^{\circ} \).<br /><br />Using Malus's law, the transmitted intensity after passing through the second polarizer is:<br /><br />\[ I = I_1 \cdot \cos^2(63.0^{\circ}) \]<br /><br />Substituting the expression for \( I_1 \), we get:<br /><br />\[ I = (I_0 \cdot \cos^2(21.0^{\circ})) \cdot \cos^2(63.0^{\circ}) \]<br /><br />Therefore, the transmitted intensity is:<br /><br />\[ I = I_0 \cdot \cos^2(21.0^{\circ}) \cdot \cos^2(63.0^{\circ}) \]<br /><br />This is the transmitted intensity of the light after passing through both polarizers.