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in q5, the second part of the question. How do we even start to do this? (it's the bit about finding the field if you assume that it's part of an infinitely large flat sheet of material)

my field from the first part of the question is

[itex]E(P)=\frac{1}{4 \pi \epsilon_0} \frac{qd}{a}[/itex]

since [itex]dE_z=\frac{dq}{4 \pi \epsilon_0}{\vec{a} \cdot \vec{\hat{z}}}{a^3}=\frac{1}{4 \pi \epsilon_0} \frac{dq a \cos{\theta}}{a^3}[/itex] then i cancelled the a's and subbed [itex]\cos{\theta}=\frac{d}{a}[/itex]