R = 1
ϕ = 0
(Subscript[U, I]) = (∫_0^(2*π))(Log[
                R] - (∑_(m = 1)^10) (1/m)*((((ρ/R)))^m)*
                  Cos[m*(θ - ϕ)])*(Cos[2*θ] + 
              Sin[2*θ]) \[DifferentialD]θ
(Subscript[U, E]) = (∫_0^(2*π))(Log[ρ] - (∑_(m = 1)^10) (1/m)*((((R/ρ)))^m)*Cos[m*(θ - ϕ)])*(Cos[
                2*θ] + 
              Sin[2*θ]) \[DifferentialD]θ

\(g1 = 
      Plot[\(-\(\(π\ ρ\^2\)\/2\)\), {ρ, 0, 1}, 
        Ticks -> {{{1, "1"}, {2, "2"}}, {{\(-π\)\/2, \*""((-π)/2)""}}}, PlotStyle -> RGBColor[1, 0, 0], 
        TextStyle -> {FontSize -> 14, 
            FontWeight -> "Bold"}]\)
\(g2 = 
      Plot[\(-\(π\/\(2\ ρ\^2\)\)\), {ρ, 1, 2}, 
        Ticks -> {{{1, "1"}, {2, "2"}}, {{\(-π\), "-π"}}}, PlotStyle -> RGBColor[0, 1, 0], 
        TextStyle -> {FontSize -> 14, 
            FontWeight -> "Bold"}]\)
\(Show[g1, g2, 
      AxesLabel -> {"a", \*""\!\(V\_U\)""}]\)

(Subscript[T, I]) = (∫_0^(2*π))((1/R) + (∑_(m = 1)^10) ((ρ^m)/(R^((m + 1))))*Cos[m*(θ - ϕ)])*(Cos[
                2*θ] + 
              Sin[2*θ]) \[DifferentialD]θ
(Subscript[T, E]) = (∫_0^(2*π))(-(∑_(m = 1)^10) ((R^((m - 1)))/(ρ^m))*Cos[m*(θ - ϕ)])*(Cos[2*θ] + 
              Sin[2*θ]) \[DifferentialD]θ

\(g3 = 
      Plot[π\ ρ\^2, {ρ, 0, 1}, 
        Ticks -> {{{1, "1"}, {2, "2"}}, {{\(-π\), "-π"}, {π, "π"}}}, PlotStyle -> RGBColor[1, 0, 0], 
        TextStyle -> {FontSize -> 14, 
            FontWeight -> "Bold"}]\)
\(g4 = 
      Plot[\(-\(π\/ρ\^2\)\), {ρ, 1, 2}, 
        Ticks -> {{{1, "1"}, {2, "2"}}, {{\(-π\), "-π"}, {π, "π"}}}, PlotStyle -> RGBColor[0, 1, 0], 
        TextStyle -> {FontSize -> 14, 
            FontWeight -> "Bold"}]\)
g5 = 
      ListPlot[{{1, π}, {1, -π}}, 
        PlotJoined -> True]
\(Show[g3, g4, g5, 
      AxesLabel -> {"a", \*""\!\(V\_T\)""}]\)

(Subscript[L, ρ I]) = (∫_0^(2*π))(-(∑_(m = 1)^10) ((ρ^(m - 1))/(R^m))*Cos[m*(θ - ϕ)])*(Cos[
                2*θ] + 
              Sin[2*θ]) \[DifferentialD]θ
(Subscript[L, ρ E]) = (∫_0^(2*π))((1/ρ) + (∑_(m = 1)^10) ((R^m)/(ρ^(m + 1)))*
                  Cos[m*(θ - ϕ)])*(Cos[2*θ] + 
              Sin[2*θ]) \[DifferentialD]θ

\(g6 = 
      Plot[\(-π\)\ ρ, {ρ, 0, 1}, 
        Ticks -> {{{1, "1"}, {2, "2"}}, {{\(-π\), "-π"}, {π, "π"}}}, PlotStyle -> RGBColor[1, 0, 0], 
        TextStyle -> {FontSize -> 14, 
            FontWeight -> "Bold"}]\)
\(g7 = 
      Plot[π\/ρ\^3, {ρ, 1, 2}, 
        Ticks -> {{{1, "1"}, {2, "2"}}, {{\(-π\), "-π"}, {π, "π"}}}, PlotStyle -> RGBColor[0, 1, 0], 
        TextStyle -> {FontSize -> 14, 
            FontWeight -> "Bold"}]\)
g8 = 
      ListPlot[{{1, -π}, {1, π}}, 
        PlotJoined -> True]
\(Show[g6, g7, g8, 
      AxesLabel -> {"a", \*""\!\(V\_\(L\_ρ\)\)""}]\)

(Subscript[M, ρ])I = (∫_0^(2*π))((∑_(m = 1)^10) 
                m*((ρ^(m - 1))/(R^((m + 1))))*
                Cos[m*(θ - ϕ)])*(Cos[2*θ] + 
              Sin[2*θ]) \[DifferentialD]θ
(Subscript[M, ρ])E = (∫_0^(2*π))((∑_(m = 1)^10) m*
                ((R^((m - 1)))/(ρ^(m + 1)))*
                Cos[m*(θ - ϕ)])*(Cos[2*θ] + 
              Sin[2*θ]) \[DifferentialD]θ

\(g9 = 
      Plot[2\ π\ ρ, {ρ, 0, 1}, 
        Ticks -> {{{1, "1"}, {2, "2"}}, {{\(-π\), "-π"}, {2*π, "2π"}}}, 
        PlotStyle -> RGBColor[1, 0, 0], 
        TextStyle -> {FontSize -> 14, 
            FontWeight -> "Bold"}]\)
\(g10 = 
      Plot[\(2\ π\)\/ρ\^3, {ρ, 1, 2}, 
        Ticks -> {{{1, "1"}, {2, "2"}}, {{\(-π\), "-π"}, {π, "π"}}}, PlotStyle -> RGBColor[0, 1, 0], 
        TextStyle -> {FontSize -> 14, 
            FontWeight -> "Bold"}]\)
\(Show[g9, g10, 
      AxesLabel -> {"a", \*""\!\(V\_\(M\_ρ\)\)""}]\)

(Subscript[L, ϕ I]) = (∫_0^(2*π))((1/ρ))*(-(∑_(m = 1)^10)((((ρ/R)))^m)*
                  Sin[m*(θ - ϕ)])*(Cos[2*θ] + 
              Sin[2*θ]) \[DifferentialD]θ
(Subscript[L, ϕ E]) = (∫_0^(2*π))((1/ρ))*(-(∑_(m = 1)^10)((((R/ρ)))^m)*
                  Sin[m*(θ - ϕ)])*(Cos[2*θ] + 
              Sin[2*θ]) \[DifferentialD]θ

\(g11 = 
      Plot[\(-π\)\ ρ, {ρ, 0, 1}, 
        Ticks -> {{{1, "1"}, {2, "2"}}, {{\(-π\), "-π"}}}, PlotStyle -> RGBColor[1, 0, 0], 
        TextStyle -> {FontSize -> 14, 
            FontWeight -> "Bold"}]\)
\(g12 = 
      Plot[\(-\(π\/ρ\^3\)\), {ρ, 1, 2}, 
        Ticks -> {{{1, "1"}, {2, "2"}}, {{\(-π\), "-π"}}}, PlotStyle -> RGBColor[0, 1, 0], 
        TextStyle -> {FontSize -> 14, 
            FontWeight -> "Bold"}]\)
\(Show[g11, g12, 
      AxesLabel -> {"a", \*""\!\(V\_\(L\_ϕ\)\)""}]\)

(Subscript[M, ϕ I]) = 
      FullSimplify[(∫_0^(2*π))((1/ρ))*((∑_(m = 1)^10) m*((ρ^m)/(R^((m + 1))))*
                  Sin[m*(θ - ϕ)])*(Cos[2*θ] + 
                Sin[2*θ]) \[DifferentialD]θ]
(Subscript[M, ϕ E]) = (∫_0^(2*π))((1/ρ))*(-(∑_(m = 1)^10) m*((R^((m - 1)))/(ρ^m))*
                  Sin[m*(θ - ϕ)])*(Cos[2*θ] + 
              Sin[2*θ]) \[DifferentialD]θ

g13 = 
      Plot[2 π ρ, {ρ, 0, 1}, 
        Ticks -> {{{1, "1"}, {2, "2"}}, {{-2*π, "-2\
π"}, {2*π, "2π"}}}, PlotStyle -> RGBColor[1, 0, 0], 
        TextStyle -> {FontSize -> 14, 
            FontWeight -> "Bold"}]
g14 = 
      Plot[-((2 π)/(ρ^3)), {ρ, 1, 2}, 
        Ticks -> {{{1, "1"}, {2, "2"}}, {{-2*π, "-2\
π"}, {2*π, "2π"}}}, PlotStyle -> RGBColor[0, 1, 0], 
        TextStyle -> {FontSize -> 14, 
            FontWeight -> "Bold"}]
g15 = 
      ListPlot[{{1, 2*π}, {1, -2*π}}, 
        PlotJoined -> True]
\(Show[g13, g14, g15, 
      AxesLabel -> {"a", \*""\!\(V\_\(M\_ϕ\)\)""}]\)