A short current element has \(d=0.03 \lambda\). Calculate the radiation resistance that is obtained for each of the following current distributions: \((a)\) uniform, \(I_{0} ;(b)\) linear, \(I(z)=I_{0}(0.5 d-|z|) / 0.5 d ;(c)\) step, \(I_{0}\) for \(0<|z|<0.25 d\) and \(0.5 I_{0}\) for \(0.25 d<|z|<0.5 d\).

For the uniform distribution, the current along the short current element is constant and given as \(I(z) = I_0\).

Using the formula for radiation intensity \(U(\theta) = \frac{1}{2}\int_{-d/2}^{d/2}I^2(z) \sin^2(kz\cos\theta) dz\), where \(k = \frac{2\pi}{\lambda}\), we can calculate the radiation intensity for the uniform case: \(U(\theta) = \frac{1}{2}I_0^2\int_{-d/2}^{d/2} \sin^2\left(\frac{2\pi}{\lambda}z\cos\theta\right)dz\)

To calculate the total power radiated \(P_r\), we need to integrate the radiation intensity over the entire solid angle: \(P_r = \int_{0}^{\pi} \int_{0}^{2\pi} U(\theta) \sin\theta d\theta d\phi\) For the uniform case, this becomes: \(P_r = I_0^2 \int_{0}^{\pi} \int_{0}^{2\pi} \frac{1}{2}\int_{-d/2}^{d/2}\sin^2\left(\frac{2\pi}{\lambda}z\cos\theta\right)dz \sin\theta d\theta d\phi\)

Now, we can calculate the radiation resistance \(R_r\) using the following formula: \(R_r = \frac{2P_r}{I_0^2}\) For the uniform case, we obtain: \(R_r = \frac{2}{I_0^2} \int_{0}^{\pi}\int_{0}^{2\pi}\frac{1}{2}\int_{-d/2}^{d/2}\sin^2\left(\frac{2\pi}{\lambda}z\cos\theta\right)dz \sin\theta d\theta d\phi\) After carrying out the integrals, we find the radiation resistance for the uniform case. ##Calculation for Linear Distribution## For the linear distribution, we follow the same procedure as for the uniform distribution, replacing the current distribution function by the linear distribution \(I(z)=I_{0}(0.5 d-|z|) / 0.5 d\). ##Calculation for Step Distribution## For the step distribution, we follow the same procedure as for the uniform distribution, replacing the current distribution function by the step distribution \(I_{0}\) for \(0<|z|<0.25 d\) and \(0.5 I_{0}\) for \(0.25 d<|z|<0.5 d\). At the end of these calculations, we will have the radiation resistance for each of the given current distributions.